Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... ·...
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![Page 1: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/1.jpg)
Class XII Chapter 7 – Integrals Maths
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Exercise 7.1
Question 1:
sin 2x
Answer
The anti derivative of sin 2x is a function of x whose derivative is sin 2x.It is known that,
Therefore, the anti derivative of
Question 2:
Cos 3x
Answer
The anti derivative of cos 3x is a function of x whose derivative is cos 3x.
It is known that,
Therefore, the anti derivative of .
Question 3:
e2x
Answer
The anti derivative of e2x is the function of x whose derivative is e2x.
It is known that,
![Page 2: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/2.jpg)
Class XII Chapter 7 – Integrals Maths
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Therefore, the anti derivative of .
Question 4:
Answer
The anti derivative of is the function of x whose derivative is .
It is known that,
Therefore, the anti derivative of .
Question 5:
Answer
The anti derivative of is the function of x whose derivative is
.
It is known that,
![Page 3: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/3.jpg)
Class XII Chapter 7 – Integrals Maths
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Therefore, the anti derivative of is .
Question 6:
Answer
Question 7:
Answer
Question 8:
Answer
![Page 4: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/4.jpg)
Class XII Chapter 7 – Integrals Maths
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Question 9:
Answer
Question 10:
Answer
![Page 5: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/5.jpg)
Class XII Chapter 7 – Integrals Maths
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Question 11:
Answer
Question 12:
Answer
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Class XII Chapter 7 – Integrals Maths
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Question 13:
Answer
On dividing, we obtain
Question 14:
Answer
Question 15:
Answer
![Page 7: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/7.jpg)
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Question 16:
Answer
Question 17:
Answer
![Page 8: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/8.jpg)
Class XII Chapter 7 – Integrals Maths
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Question 18:
Answer
Question 19:
Answer
![Page 9: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/9.jpg)
Class XII Chapter 7 – Integrals Maths
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Question 20:
Answer
Question 21:
The anti derivative of equals
(A) (B)
(C) (D)
Answer
![Page 10: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/10.jpg)
Class XII Chapter 7 – Integrals Maths
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Hence, the correct Answer is C.
Question 22:
If such that f(2) = 0, then f(x) is
(A) (B)
(C) (D)
Answer
It is given that,
∴Anti derivative of
![Page 11: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/11.jpg)
Class XII Chapter 7 – Integrals Maths
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∴
Also,
Hence, the correct Answer is A.
![Page 12: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/12.jpg)
Class XII Chapter 7 – Integrals Maths
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Exercise 7.2
Question 1:
Answer
Let = t
∴2x dx = dt
Question 2:
Answer
Let log |x| = t
∴
![Page 13: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/13.jpg)
Class XII Chapter 7 – Integrals Maths
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Question 3:
Answer
Let 1 + log x = t
∴
Question 4:
sin x ⋅ sin (cos x)
Answer
![Page 14: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/14.jpg)
Class XII Chapter 7 – Integrals Maths
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sin x ⋅ sin (cos x)
Let cos x = t
∴ −sin x dx = dt
Question 5:
Answer
Let
∴ 2adx = dt
![Page 15: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/15.jpg)
Class XII Chapter 7 – Integrals Maths
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Question 6:
Answer
Let ax + b = t
⇒ adx = dt
Question 7:
Answer
Let
![Page 16: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/16.jpg)
Class XII Chapter 7 – Integrals Maths
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∴ dx = dt
Question 8:
Answer
Let 1 + 2x2 = t
∴ 4xdx = dt
![Page 17: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/17.jpg)
Class XII Chapter 7 – Integrals Maths
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Question 9:
Answer
Let
∴ (2x + 1)dx = dt
![Page 18: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/18.jpg)
Class XII Chapter 7 – Integrals Maths
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Question 10:
Answer
Let
∴
Question 11:
Answer
Let
∴ dx = dt
![Page 19: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/19.jpg)
Class XII Chapter 7 – Integrals Maths
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Question 12:
Answer
Let
∴
![Page 20: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/20.jpg)
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Question 13:
Answer
Let
∴ 9x2 dx = dt
![Page 21: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/21.jpg)
Class XII Chapter 7 – Integrals Maths
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Question 14:
Answer
Let log x = t
∴
Question 15:
Answer
![Page 22: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/22.jpg)
Class XII Chapter 7 – Integrals Maths
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Let
∴ −8x dx = dt
Question 16:
Answer
Let
∴ 2dx = dt
![Page 23: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/23.jpg)
Class XII Chapter 7 – Integrals Maths
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Question 17:
Answer
Let
∴ 2xdx = dt
Question 18:
Answer
Let
∴
![Page 24: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/24.jpg)
Class XII Chapter 7 – Integrals Maths
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Question 19:
Answer
Dividing numerator and denominator by ex, we obtain
Let
∴
![Page 25: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/25.jpg)
Class XII Chapter 7 – Integrals Maths
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Question 20:
Answer
Let
∴
Question 21:
Answer
Let 2x − 3 = t
∴ 2dx = dt
![Page 26: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/26.jpg)
Class XII Chapter 7 – Integrals Maths
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Question 22:
Answer
Let 7 − 4x = t
∴ −4dx = dt
Question 23:
Answer
Let
![Page 27: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/27.jpg)
Class XII Chapter 7 – Integrals Maths
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∴
Question 24:
Answer
Let
∴
![Page 28: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/28.jpg)
Class XII Chapter 7 – Integrals Maths
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Question 25:
Answer
Let
∴
Question 26:
Answer
Let
∴
![Page 29: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/29.jpg)
Class XII Chapter 7 – Integrals Maths
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Question 27:
Answer
Let sin 2x = t
∴
Question 28:
Answer
Let
![Page 30: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/30.jpg)
Class XII Chapter 7 – Integrals Maths
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∴ cos x dx = dt
Question 29:
cot x log sin x
Answer
Let log sin x = t
Question 30:
Answer
Let 1 + cos x = t
![Page 31: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/31.jpg)
Class XII Chapter 7 – Integrals Maths
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∴ −sin x dx = dt
Question 31:
Answer
Let 1 + cos x = t
∴ −sin x dx = dt
Question 32:
Answer
![Page 32: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/32.jpg)
Class XII Chapter 7 – Integrals Maths
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Let sin x + cos x = t ⇒ (cos x − sin x) dx = dt
Question 33:
Answer
![Page 33: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/33.jpg)
Class XII Chapter 7 – Integrals Maths
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Put cos x − sin x = t ⇒ (−sin x − cos x) dx = dt
Question 34:
Answer
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Class XII Chapter 7 – Integrals Maths
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Question 35:
Answer
Let 1 + log x = t
∴
![Page 35: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/35.jpg)
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Question 36:
Answer
Let
∴
Question 37:
Answer
Let x4 = t
∴ 4x3 dx = dt
![Page 36: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/36.jpg)
Class XII Chapter 7 – Integrals Maths
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Let
∴
From (1), we obtain
Question 38:
equals
Answer
Let
∴
![Page 37: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/37.jpg)
Class XII Chapter 7 – Integrals Maths
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Hence, the correct Answer is D.
Question 39:
equals
A.
B.
C.
D.
Answer
Hence, the correct Answer is B.
![Page 38: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/38.jpg)
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Exercise 7.3
Question 1:
Answer
Question 2:
Answer
It is known that,
![Page 39: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/39.jpg)
Class XII Chapter 7 – Integrals Maths
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Question 3:
cos 2x cos 4x cos 6x
Answer
It is known that,
Question 4:
sin3 (2x + 1)
Answer
Let
![Page 40: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/40.jpg)
Class XII Chapter 7 – Integrals Maths
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Question 5:
sin3 x cos3 x
Answer
Question 6:
sin x sin 2x sin 3x
Answer
![Page 41: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/41.jpg)
Class XII Chapter 7 – Integrals Maths
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It is known that,
Question 7:
sin 4x sin 8x
Answer
It is known that,
![Page 42: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/42.jpg)
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Question 8:
Answer
Question 9:
Answer
![Page 43: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/43.jpg)
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Question 10:
sin4 x
Answer
![Page 44: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/44.jpg)
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Question 11:
cos4 2x
Answer
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Question 12:
Answer
![Page 46: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/46.jpg)
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Question 13:
Answer
Question 14:
Answer
![Page 47: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/47.jpg)
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Question 15:
Answer
![Page 48: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/48.jpg)
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Question 16:
tan4x
Answer
![Page 49: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/49.jpg)
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From equation (1), we obtain
Question 17:
Answer
Question 18:
Answer
Question 19:
![Page 50: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/50.jpg)
Class XII Chapter 7 – Integrals Maths
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Answer
Question 20:
Answer
![Page 51: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/51.jpg)
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Question 21:
sin−1 (cos x)
Answer
![Page 52: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/52.jpg)
Class XII Chapter 7 – Integrals Maths
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It is known that,
Substituting in equation (1), we obtain
![Page 53: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/53.jpg)
Class XII Chapter 7 – Integrals Maths
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Question 22:
Answer
![Page 54: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/54.jpg)
Class XII Chapter 7 – Integrals Maths
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Question 23:
is equal to
A. tan x + cot x + C
B. tan x + cosec x + C
C. − tan x + cot x + C
D. tan x + sec x + C
Answer
Hence, the correct Answer is A.
Question 24:
equals
A. − cot (exx) + C
B. tan (xex) + C
C. tan (ex) + C
D. cot (ex) + C
Answer
Let exx = t
![Page 55: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/55.jpg)
Class XII Chapter 7 – Integrals Maths
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Hence, the correct Answer is B.
![Page 56: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/56.jpg)
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Exercise 7.4
Question 1:
Answer
Let x3 = t
∴ 3x2 dx = dt
Question 2:
Answer
Let 2x = t
∴ 2dx = dt
![Page 57: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/57.jpg)
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Question 3:
Answer
Let 2 − x = t
⇒ −dx = dt
Question 4:
Answer
Let 5x = t
![Page 58: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/58.jpg)
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∴ 5dx = dt
Question 5:
Answer
Question 6:
Answer
Let x3 = t
![Page 59: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/59.jpg)
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∴ 3x2 dx = dt
Question 7:
Answer
From (1), we obtain
![Page 60: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/60.jpg)
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Question 8:
Answer
Let x3 = t
⇒ 3x2 dx = dt
Question 9:
Answer
Let tan x = t
∴ sec2x dx = dt
![Page 61: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/61.jpg)
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Question 10:
Answer
Question 11:
Answer
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Question 12:
Answer
![Page 63: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/63.jpg)
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Question 13:
Answer
Question 14:
![Page 64: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/64.jpg)
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Answer
Question 15:
Answer
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Question 16:
Answer
Equating the coefficients of x and constant term on both sides, we obtain
![Page 66: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/66.jpg)
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4A = 4 ⇒ A = 1
A + B = 1 ⇒ B = 0
Let 2x2 + x − 3 = t
∴ (4x + 1) dx = dt
Question 17:
Answer
Equating the coefficients of x and constant term on both sides, we obtain
From (1), we obtain
![Page 67: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/67.jpg)
Class XII Chapter 7 – Integrals Maths
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From equation (2), we obtain
Question 18:
Answer
Equating the coefficient of x and constant term on both sides, we obtain
![Page 68: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/68.jpg)
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![Page 69: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/69.jpg)
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Substituting equations (2) and (3) in equation (1), we obtain
![Page 70: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/70.jpg)
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Question 19:
Answer
Equating the coefficients of x and constant term, we obtain
2A = 6 ⇒ A = 3
−9A + B = 7 ⇒ B = 34
∴ 6x + 7 = 3 (2x − 9) + 34
![Page 71: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/71.jpg)
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Substituting equations (2) and (3) in (1), we obtain
Question 20:
Answer
Equating the coefficients of x and constant term on both sides, we obtain
![Page 73: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/73.jpg)
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Using equations (2) and (3) in (1), we obtain
![Page 74: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/74.jpg)
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Question 21:
Answer
Let x2 + 2x +3 = t
⇒ (2x + 2) dx =dt
![Page 75: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/75.jpg)
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Using equations (2) and (3) in (1), we obtain
Question 22:
Answer
Equating the coefficients of x and constant term on both sides, we obtain
![Page 76: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/76.jpg)
Class XII Chapter 7 – Integrals Maths
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Substituting (2) and (3) in (1), we obtain
Question 23:
Answer
![Page 77: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/77.jpg)
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Equating the coefficients of x and constant term, we obtain
Using equations (2) and (3) in (1), we obtain
![Page 78: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/78.jpg)
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Question 24:
equals
A. x tan−1 (x + 1) + C
B. tan− 1 (x + 1) + C
C. (x + 1) tan−1 x + C
D. tan−1 x + C
Answer
Hence, the correct Answer is B.
Question 25:
equals
A.
B.
C.
![Page 79: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/79.jpg)
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D.
Answer
Hence, the correct Answer is B.
![Page 80: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/80.jpg)
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Exercise 7.5
Question 1:
Answer
Let
Equating the coefficients of x and constant term, we obtain
A + B = 1
2A + B = 0
On solving, we obtain
A = −1 and B = 2
Question 2:
Answer
Let
![Page 81: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/81.jpg)
Class XII Chapter 7 – Integrals Maths
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Equating the coefficients of x and constant term, we obtain
A + B = 0
−3A + 3B = 1
On solving, we obtain
Question 3:
Answer
Let
Substituting x = 1, 2, and 3 respectively in equation (1), we obtain
A = 1, B = −5, and C = 4
![Page 82: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/82.jpg)
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Question 4:
Answer
Let
Substituting x = 1, 2, and 3 respectively in equation (1), we obtain
Question 5:
Answer
Let
Substituting x = −1 and −2 in equation (1), we obtain
A = −2 and B = 4
![Page 83: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/83.jpg)
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Question 6:
Answer
It can be seen that the given integrand is not a proper fraction.
Therefore, on dividing (1 − x2) by x(1 − 2x), we obtain
Let
Substituting x = 0 and in equation (1), we obtain
A = 2 and B = 3
Substituting in equation (1), we obtain
![Page 84: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/84.jpg)
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Question 7:
Answer
Let
Equating the coefficients of x2, x, and constant term, we obtain
A + C = 0
−A + B = 1
−B + C = 0
On solving these equations, we obtain
From equation (1), we obtain
![Page 85: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/85.jpg)
Class XII Chapter 7 – Integrals Maths
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Question 8:
Answer
Let
Substituting x = 1, we obtain
Equating the coefficients of x2 and constant term, we obtain
A + C = 0
−2A + 2B + C = 0
On solving, we obtain
![Page 86: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/86.jpg)
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Question 9:
Answer
Let
Substituting x = 1 in equation (1), we obtain
B = 4
Equating the coefficients of x2 and x, we obtain
A + C = 0
B − 2C = 3
On solving, we obtain
![Page 87: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/87.jpg)
Class XII Chapter 7 – Integrals Maths
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Question 10:
Answer
Let
Equating the coefficients of x2 and x, we obtain
![Page 88: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/88.jpg)
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Question 11:
Answer
Let
Substituting x = −1, −2, and 2 respectively in equation (1), we obtain
Question 12:
![Page 89: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/89.jpg)
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Answer
It can be seen that the given integrand is not a proper fraction.
Therefore, on dividing (x3 + x + 1) by x2 − 1, we obtain
Let
Substituting x = 1 and −1 in equation (1), we obtain
Question 13:
Answer
Equating the coefficient of x2, x, and constant term, we obtain
A − B = 0
B − C = 0
A + C = 2
![Page 90: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/90.jpg)
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On solving these equations, we obtain
A = 1, B = 1, and C = 1
Question 14:
Answer
Equating the coefficient of x and constant term, we obtain
A = 3
2A + B = −1 ⇒ B = −7
![Page 91: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/91.jpg)
Class XII Chapter 7 – Integrals Maths
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Question 15:
Answer
Equating the coefficient of x3, x2, x, and constant term, we obtain
On solving these equations, we obtain
![Page 92: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/92.jpg)
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Question 16:
[Hint: multiply numerator and denominator by xn − 1 and put xn = t]
Answer
Multiplying numerator and denominator by xn − 1, we obtain
Substituting t = 0, −1 in equation (1), we obtain
A = 1 and B = −1
![Page 93: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/93.jpg)
Class XII Chapter 7 – Integrals Maths
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Question 17:
[Hint: Put sin x = t]
Answer
Substituting t = 2 and then t = 1 in equation (1), we obtain
A = 1 and B = −1
![Page 94: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/94.jpg)
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Question 18:
Answer
Equating the coefficients of x3, x2, x, and constant term, we obtain
A + C = 0
B + D = 4
4A + 3C = 0
4B + 3D = 10
On solving these equations, we obtain
A = 0, B = −2, C = 0, and D = 6
![Page 95: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/95.jpg)
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Question 19:
Answer
Let x2 = t ⇒ 2x dx = dt
Substituting t = −3 and t = −1 in equation (1), we obtain
![Page 96: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/96.jpg)
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Question 20:
Answer
Multiplying numerator and denominator by x3, we obtain
Let x4 = t ⇒ 4x3dx = dt
![Page 97: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/97.jpg)
Class XII Chapter 7 – Integrals Maths
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Substituting t = 0 and 1 in (1), we obtain
A = −1 and B = 1
Question 21:
[Hint: Put ex = t]
Answer
Let ex = t ⇒ ex dx = dt
![Page 98: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/98.jpg)
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Substituting t = 1 and t = 0 in equation (1), we obtain
A = −1 and B = 1
Question 22:
A.
B.
C.
D.
Answer
Substituting x = 1 and 2 in (1), we obtain
A = −1 and B = 2
![Page 99: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/99.jpg)
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Hence, the correct Answer is B.
Question 23:
A.
B.
C.
D.
Answer
Equating the coefficients of x2, x, and constant term, we obtain
A + B = 0
C = 0
A = 1
On solving these equations, we obtain
A = 1, B = −1, and C = 0
![Page 100: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/100.jpg)
Class XII Chapter 7 – Integrals Maths
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Hence, the correct Answer is A.
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Exercise 7.6
Question 1:
x sin x
Answer
Let I =
Taking x as first function and sin x as second function and integrating by parts, we
obtain
Question 2:
Answer
Let I =
Taking x as first function and sin 3x as second function and integrating by parts, we
obtain
Question 3:
![Page 102: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/102.jpg)
Class XII Chapter 7 – Integrals Maths
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Answer
Let
Taking x2 as first function and ex as second function and integrating by parts, we obtain
Again integrating by parts, we obtain
Question 4:
x logx
Answer
Let
Taking log x as first function and x as second function and integrating by parts, we
obtain
![Page 103: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/103.jpg)
Class XII Chapter 7 – Integrals Maths
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Question 5:
x log 2x
Answer
Let
Taking log 2x as first function and x as second function and integrating by parts, we
obtain
Question 6:
x2 log x
Answer
Let
Taking log x as first function and x2 as second function and integrating by parts, we
obtain
![Page 104: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/104.jpg)
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Question 7:
Answer
Let
Taking as first function and x as second function and integrating by parts, we
obtain
Question 8:
Answer
Let
![Page 105: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/105.jpg)
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Taking as first function and x as second function and integrating by parts, we
obtain
Question 9:
Answer
Let
Taking cos−1 x as first function and x as second function and integrating by parts, we
obtain
![Page 106: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/106.jpg)
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Question 10:
Answer
Let
Taking as first function and 1 as second function and integrating by parts, we
obtain
Question 11:
Answer
Let
![Page 108: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/108.jpg)
Class XII Chapter 7 – Integrals Maths
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Taking as first function and as second function and integrating by parts,
we obtain
Question 12:
Answer
Let
Taking x as first function and sec2x as second function and integrating by parts, we
obtain
Question 13:
Answer
![Page 109: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/109.jpg)
Class XII Chapter 7 – Integrals Maths
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Let
Taking as first function and 1 as second function and integrating by parts, we
obtain
Question 14:
Answer
Taking as first function and 1 as second function and integrating by parts, we
obtain
Again integrating by parts, we obtain
![Page 110: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/110.jpg)
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Question 15:
Answer
Let
Let I = I1 + I2 … (1)
Where, and
Taking log x as first function and x2 as second function and integrating by parts, we
obtain
Taking log x as first function and 1 as second function and integrating by parts, we
obtain
![Page 111: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/111.jpg)
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Using equations (2) and (3) in (1), we obtain
Question 16:
Answer
Let
Let
⇒
⇒
It is known that,
Question 17:
![Page 112: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/112.jpg)
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Answer
Let
Let ⇒
It is known that,
Question 18:
Answer
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Let ⇒
It is known that,
From equation (1), we obtain
Question 19:
![Page 114: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/114.jpg)
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Answer
Also, let ⇒
It is known that,
Question 20:
Answer
Let ⇒
It is known that,
Question 21:
Answer
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Let
Integrating by parts, we obtain
Again integrating by parts, we obtain
Question 22:
Answer
Let ⇒
![Page 116: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/116.jpg)
Class XII Chapter 7 – Integrals Maths
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= 2θ
⇒
Integrating by parts, we obtain
Question 23:
equals
Answer
Let
Also, let ⇒
![Page 117: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/117.jpg)
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Hence, the correct Answer is A.
Question 24:
equals
Answer
Let
Also, let ⇒
It is known that,
Hence, the correct Answer is B.
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Exercise 7.7
Question 1:
Answer
Question 2:
Answer
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Question 3:
Answer
Question 4:
Answer
Question 5:
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Answer
Question 6:
Answer
Question 7:
Answer
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Question 8:
Answer
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Question 9:
Answer
Question 10:
is equal to
A.
B.
C.
![Page 123: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/123.jpg)
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D.
Answer
Hence, the correct Answer is A.
Question 11:
is equal to
A.
B.
C.
D.
Answer
Hence, the correct Answer is D.
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Exercise 7.8
Question 1:
Answer
It is known that,
![Page 125: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/125.jpg)
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Question 2:
Answer
It is known that,
Question 3:
![Page 126: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/126.jpg)
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Answer
It is known that,
![Page 127: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/127.jpg)
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Question 4:
Answer
It is known that,
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From equations (2) and (3), we obtain
![Page 129: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/129.jpg)
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Question 5:
Answer
It is known that,
![Page 130: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/130.jpg)
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Question 6:
Answer
It is known that,
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Exercise 7.9
Question 1:
Answer
By second fundamental theorem of calculus, we obtain
Question 2:
Answer
By second fundamental theorem of calculus, we obtain
Question 3:
![Page 133: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/133.jpg)
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Answer
By second fundamental theorem of calculus, we obtain
Question 4:
Answer
By second fundamental theorem of calculus, we obtain
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Question 5:
Answer
By second fundamental theorem of calculus, we obtain
Question 6:
Answer
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By second fundamental theorem of calculus, we obtain
Question 7:
Answer
By second fundamental theorem of calculus, we obtain
Question 8:
Answer
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By second fundamental theorem of calculus, we obtain
Question 9:
Answer
By second fundamental theorem of calculus, we obtain
Question 10:
![Page 137: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/137.jpg)
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Answer
By second fundamental theorem of calculus, we obtain
Question 11:
Answer
By second fundamental theorem of calculus, we obtain
Question 12:
![Page 138: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/138.jpg)
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Answer
By second fundamental theorem of calculus, we obtain
Question 13:
Answer
By second fundamental theorem of calculus, we obtain
![Page 139: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/139.jpg)
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Question 14:
Answer
By second fundamental theorem of calculus, we obtain
Question 15:
Answer
![Page 140: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/140.jpg)
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By second fundamental theorem of calculus, we obtain
Question 16:
Answer
Let
![Page 141: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/141.jpg)
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Equating the coefficients of x and constant term, we obtain
A = 10 and B = −25
Substituting the value of I1 in (1), we obtain
![Page 142: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/142.jpg)
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Question 17:
Answer
By second fundamental theorem of calculus, we obtain
Question 18:
Answer
![Page 143: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/143.jpg)
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By second fundamental theorem of calculus, we obtain
Question 19:
Answer
By second fundamental theorem of calculus, we obtain
![Page 144: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/144.jpg)
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Question 20:
Answer
By second fundamental theorem of calculus, we obtain
![Page 145: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/145.jpg)
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Question 21:
equals
A.
B.
C.
D.
Answer
By second fundamental theorem of calculus, we obtain
Hence, the correct Answer is D.
![Page 146: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/146.jpg)
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Question 22:
equals
A.
B.
C.
D.
Answer
By second fundamental theorem of calculus, we obtain
![Page 147: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/147.jpg)
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Hence, the correct Answer is C.
![Page 148: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/148.jpg)
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Exercise 7.10
Question 1:
Answer
When x = 0, t = 1 and when x = 1, t = 2
Question 2:
Answer
Also, let
![Page 149: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/149.jpg)
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Question 3:
Answer
Also, let x = tanθ ⇒ dx = sec2θ dθ
When x = 0, θ = 0 and when x = 1,
![Page 150: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/150.jpg)
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Takingθas first function and sec2θ as second function and integrating by parts, we obtain
Question 4:
Answer
Let x + 2 = t2 ⇒ dx = 2tdt
When x = 0, and when x = 2, t = 2
![Page 151: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/151.jpg)
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Question 5:
Answer
Let cos x = t ⇒ −sinx dx = dt
When x = 0, t = 1 and when
![Page 152: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/152.jpg)
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Question 6:
Answer
Let ⇒ dx = dt
![Page 153: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/153.jpg)
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Question 7:
Answer
Let x + 1 = t ⇒ dx = dt
When x = −1, t = 0 and when x = 1, t = 2
Question 8:
Answer
Let 2x = t ⇒ 2dx = dt
When x = 1, t = 2 and when x = 2, t = 4
![Page 155: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/155.jpg)
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Question 9:
The value of the integral is
A. 6
B. 0
C. 3
D. 4
Answer
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Let cotθ = t ⇒ −cosec2θ dθ= dt
![Page 157: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/157.jpg)
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Hence, the correct Answer is A.
Question 10:
If
A. cos x + x sin x
B. x sin x
C. x cos x
D. sin x + x cos x
Answer
Integrating by parts, we obtain
![Page 158: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/158.jpg)
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Hence, the correct Answer is B.
![Page 159: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/159.jpg)
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Exercise 7.11
Question 1:
Answer
Adding (1) and (2), we obtain
Question 2:
Answer
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Adding (1) and (2), we obtain
Question 3:
Answer
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Adding (1) and (2), we obtain
Question 4:
Answer
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Adding (1) and (2), we obtain
Question 5:
Answer
It can be seen that (x + 2) ≤ 0 on [−5, −2] and (x + 2) ≥ 0 on [−2, 5].
![Page 163: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/163.jpg)
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Question 6:
Answer
It can be seen that (x − 5) ≤ 0 on [2, 5] and (x − 5) ≥ 0 on [5, 8].
Question 7:
Answer
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Question 8:
Answer
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Question 9:
Answer
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Question 10:
Answer
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Adding (1) and (2), we obtain
Question 11:
Answer
As sin2 (−x) = (sin (−x))2 = (−sin x)2 = sin2x, therefore, sin2x is an even function.
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It is known that if f(x) is an even function, then
Question 12:
Answer
Adding (1) and (2), we obtain
![Page 169: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/169.jpg)
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Question 13:
Answer
As sin7 (−x) = (sin (−x))7 = (−sin x)7 = −sin7x, therefore, sin2x is an odd function.
It is known that, if f(x) is an odd function, then
Question 14:
Answer
It is known that,
![Page 170: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/170.jpg)
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Question 15:
Answer
Adding (1) and (2), we obtain
Question 16:
Answer
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Adding (1) and (2), we obtain
sin (π − x) = sin x
Adding (4) and (5), we obtain
Let 2x = t ⇒ 2dx = dt
When x = 0, t = 0 and when
![Page 172: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/172.jpg)
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Question 17:
Answer
It is known that,
Adding (1) and (2), we obtain
Question 18:
Answer
It can be seen that, (x − 1) ≤ 0 when 0 ≤ x ≤ 1 and (x − 1) ≥ 0 when 1 ≤ x ≤ 4
![Page 173: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/173.jpg)
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Question 19:
Show that if f and g are defined as and
Answer
Adding (1) and (2), we obtain
Question 20:
![Page 174: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/174.jpg)
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The value of is
A. 0
B. 2
C. π
D. 1
Answer
It is known that if f(x) is an even function, then and
if f(x) is an odd function, then
Hence, the correct Answer is C.
Question 21:
The value of is
A. 2
B.
C. 0
D.
![Page 175: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/175.jpg)
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Answer
Adding (1) and (2), we obtain
Hence, the correct Answer is C.
![Page 176: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/176.jpg)
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Miscellaneous Solutions
Question 1:
Answer
Equating the coefficients of x2, x, and constant term, we obtain
−A + B − C = 0
B + C = 0
A = 1
On solving these equations, we obtain
From equation (1), we obtain
![Page 177: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/177.jpg)
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Question 2:
Answer
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Question 3:
[Hint: Put ]
Answer
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Question 4:
Answer
![Page 180: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/180.jpg)
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Question 5:
Answer
On dividing, we obtain
Question 6:
![Page 182: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/182.jpg)
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Answer
Equating the coefficients of x2, x, and constant term, we obtain
A + B = 0
B + C = 5
9A + C = 0
On solving these equations, we obtain
From equation (1), we obtain
Question 7:
Answer
![Page 183: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/183.jpg)
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Let x − a = t ⇒ dx = dt
Question 8:
Answer
Question 9:
Answer
![Page 184: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/184.jpg)
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Let sin x = t ⇒ cos x dx = dt
Question 10:
Answer
Question 11:
Answer
![Page 185: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/185.jpg)
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Question 12:
Answer
Let x4 = t ⇒ 4x3 dx = dt
![Page 186: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/186.jpg)
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Question 13:
Answer
Let ex = t ⇒ ex dx = dt
Question 14:
Answer
![Page 187: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/187.jpg)
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Equating the coefficients of x3, x2, x, and constant term, we obtain
A + C = 0
B + D = 0
4A + C = 0
4B + D = 1
On solving these equations, we obtain
From equation (1), we obtain
Question 15:
Answer
= cos3 x × sin x
Let cos x = t ⇒ −sin x dx = dt
![Page 188: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/188.jpg)
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Question 16:
Answer
Question 17:
Answer
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Question 18:
Answer
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Question 19:
Answer
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From equation (1), we obtain
![Page 192: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/192.jpg)
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Question 20:
Answer
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Question 21:
Answer
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Question 22:
Answer
Equating the coefficients of x2, x,and constant term, we obtain
A + C = 1
3A + B + 2C = 1
2A + 2B + C = 1
On solving these equations, we obtain
A = −2, B = 1, and C = 3
From equation (1), we obtain
![Page 195: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/195.jpg)
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Question 23:
Answer
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Question 24:
Answer
Integrating by parts, we obtain
![Page 197: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/197.jpg)
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Question 25:
Answer
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Question 26:
Answer
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When x = 0, t = 0 and
Question 27:
Answer
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When and when
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Question 28:
Answer
When and when
As , therefore, is an even function.
It is known that if f(x) is an even function, then
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Question 29:
Answer
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Question 30:
Answer
Question 31:
Answer
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From equation (1), we obtain
Question 32:
Answer
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Adding (1) and (2), we obtain
Question 33:
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Answer
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From equations (1), (2), (3), and (4), we obtain
Question 34:
Answer
Equating the coefficients of x2, x, and constant term, we obtain
A + C = 0
A + B = 0
B = 1
On solving these equations, we obtain
A = −1, C = 1, and B = 1
![Page 208: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/208.jpg)
Class XII Chapter 7 – Integrals Maths
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Hence, the given result is proved.
Question 35:
Answer
Integrating by parts, we obtain
![Page 209: Chapter 7 Integrals - Mathshelp4Umathshelp4u.weebly.com/uploads/1/6/0/1/16016512/chapter... · 2019. 5. 12. · Class XII Chapter 7 – Integrals Maths Page 4 of 216 Website: Email:](https://reader030.fdocuments.in/reader030/viewer/2022013010/600f360bdc12fb73313675a3/html5/thumbnails/209.jpg)
Class XII Chapter 7 – Integrals Maths
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Hence, the given result is proved.
Question 36:
Answer
Therefore, f (x) is an odd function.
It is known that if f(x) is an odd function, then
Hence, the given result is proved.
Question 37:
Answer
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Class XII Chapter 7 – Integrals Maths
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Hence, the given result is proved.
Question 38:
Answer
Hence, the given result is proved.
Question 39:
Answer
Integrating by parts, we obtain
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Class XII Chapter 7 – Integrals Maths
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Let 1 − x2 = t ⇒ −2x dx = dt
Hence, the given result is proved.
Question 40:
Evaluate as a limit of a sum.
Answer
It is known that,
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Question 41:
is equal to
A.
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Class XII Chapter 7 – Integrals Maths
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B.
C.
D.
Answer
Hence, the correct Answer is A.
Question 42:
is equal to
A.
B.
C.
D.
Answer
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Class XII Chapter 7 – Integrals Maths
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Hence, the correct Answer is B.
Question 43:
If then is equal to
A.
B.
C.
D.
Answer
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Class XII Chapter 7 – Integrals Maths
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Hence, the correct Answer is D.
Question 44:
The value of is
A. 1
B. 0
C. − 1
D.
Answer
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Class XII Chapter 7 – Integrals Maths
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Adding (1) and (2), we obtain
Hence, the correct Answer is B.