INDEFINITE INTEGRATION By Prof. Rajnish Mishra · INDEFINITE INTEGRATION By Prof. Rajnish Mishra 2...

INDEFINITE INTEGRATION By Prof. Rajnish Mishra

1

Exercise-1

1. cos xxe x dx is equal to

a. a=1,b=1, c=-1

b. a=1

, 1, 12

b c

c. 1, 1, 1a b c

d. 1

, 1, 12

a b c

2. 2sin nx dx is equal to

a. 5 2sin 2 cos 210

xnx nx C

b. 5 2sin 2 cos 210

xnx nx C

c. 5 2sin 210

xnx C

d. 5 2sin 2 cos 210

xnx nx C

3.

22

3

2

2

1

x xdx

x

is equal to

a. 4 6 2

23 11

4 6 2 2

x x xn x C

b. 4 6 2

23 11

4 6 2 2

x x xn x C

c. 4 6 2

23 11

4 6 2 2

x x xn x C

d. 4 6 2

23 11

4 6 2 2

x x xn x C

4. sin 1 .cos 2

dx

x x is equal to

a.

sin 1cos1.

sec 1

xn C

x

b.

sin 1sec1.

sec 1

xn C

x


2

c. cos1. sin 1 .sec 2n x x C

d. sec1. sin 1 .sec 2n x x C

5. 2

2 2x n nx C is equal to

a.

sinsin b-a .

sin

x an C

x a

b.

sincos .

sin

x aec b a n C

x b

c. 2x nx 2 2nx C d. 2

x nx 2 2nx C

6. 3 1tanx d x

is equal to

a. 2

211

2 2

xn x C b.

221

12 2

xn x C

c. 1tanx x x C d. 11 tanx x x C

7.

2

4 2

1

1

xdx

x x

is equal to

a.

2

1

2

1 1tan

3

xC

x

b.

2

1

2

1 1tan

3 3

x

x

c. 2

11 1tan

3 3

xC

x

d. 2

11 1tan

3 3

xC

8. 2

2sin

3 sin

xdx

x is equal to

a. 11 2 sin cos 1 sin cos

tan2 2 sin cos 2 3

x x x xn

x x

b. 11 2 sin cos 1 sin cos

tan2 2 sin cos 2 2 2

x x x xn C

x x

c. 11 2 sin cos 1 sin cos

tan4 2 sin cos 2 2

x x x xn C

x x

d. None of these

9.

2

2

1

1

x xe

x

a. 21

xeC

x

b.

21

xeC

x


3

c.

2

21

xeC

x

d.

2

21

xeC

x

10. 2

1x xe dx

x

a.

xeC

x b.

xeC

x c.

2

xeC

x d.

2

xeC

x

11.

9

24 1

xdx

x is equal to

a.

5

2

1 14

5C

x x

b.

5

2

1 14

5C

x

c.

521

1 410

x Cx

d. 5

211 4

10x C

12.

3

1

1

x xe dx

x

is equal to

a.

3/ 21

xeC

x

b.

2

1

xeC

x

c.

3/ 2

1

xeC

x

d.

21

xeC

x

13. If

2 1

2

11 tan 2

52 1

dxa n x b x n x C

x x

then

a. 1 2

,10 5

a b b. 1 2

,10 5

a b c. 1 2

,10 5

a b

d. 1 2

,10 5

a b

14. sin

sin 4

xdx

x is equal to

a. 1 1 2 sin 1 1 sin

8 1 sin2 2 1 2 sin

x xn n C

xx

b. 1 1 2 sin 1 1 sin

8 1 sin2 2 1 2 sin

x xn n C

xx


4

c. 1 1 2 sin 1 1 sin

8 1 sin4 2 1 2 sin

x xn n C

xx

d. 1 1 2 sin 1 1 sin

8 1 sin4 2 1 2 sin

x xn n C

xx

15. 2/3 2/31

dx

x x is equal to

a. 1 1/31tan

3x C b. 1 1/33tan x C c. 1 1/317

sin88

x C d. 1 1/33sin x C

16. 2 22

dx

x x is equal to

a. 21x x C b. 211 x C

x c. 21

1 x Cx

d. 211 x C

x

17. If 3sin 2cos

2sin 3cos3cos 2sin

x xdx ax b n x x C

x x

then

a. 12 15

,13 39

a b b. 17 6

,13 13

a b

c. 12 15

,13 39

a b d. 17 1

,13 192

a b

18. If 2 4sin .cos sin 2 sin 4 sin6x xdx ax b x c x d x e then

a. 1 1 1 1

, , ,16 32 64 192

a b c d

b. 1 1 1 1

, , ,16 32 64 192

a b c d c.

/ 2

0

1 sin

2

xdx

x

d.

/ 2

0

1

2 sin

xdx

x

19. 1

1

xdx

x x

is equal to

a. 2 11 tann x x C b.

2 11 tann x x C c. 2 11 secn x x C

d. 2 11 secn x x C

20.

21

dx

x x x is equal to

a. 2 1

1

xC

x

b.

2 1

1

xC

x

c.

2 1

1

xC

x

d.

2 1

1

xC

x


5

21. 6 6sin cos

dx

x x is equal to

a. tan cotn x x C

b. tan cotn x x C

c. 1tan 2cot 2x C

d. 1tan tan cotx x C

22. 8 13

4 7

xdx

x

is equal to

a. 1

8 11 4 76

x x C b. 1

8 13 4 76

x x C c. 1

8 9 4 76

x x C

d. 1

8 15 4 76

x x C

23.

3/ 2 2

2

1 1xx dx

x x

is equal to

a.

31 1

3x C

x

b.

5/ 22 1

5x C

x

c.

3/ 22 1

3x C

x

d.

3/ 44 1

3x C

x

24.

2

1

xdx

x is equal to

a.

2 2

12 3

x xx n x C

b.

2 2

12 3

x xx n x C

c.

2 2

12 3

x xx n x C

d.

2 2

12 3

x xx n x C

25. sin

1 cos

x xdx

x

is equal to

a. 1 cosn x C b. sinn x x C c. tan2

xx C d. . tan

2

xx C


6

Exercise-2

1. If cos 1 sin cosx xxe xdx ae b x x cx x d then

a. a=1, b. b=1,c=-1

b. 1

, 1, 12

a b c

c. 1, 1, 1a b c

d. a=1

, 1, 12

b c

2. 2sin nx dx is equal to

a. 5 2sin 2 cos 210

xnx nx C

b. 5 2sin 2 cos 210

xnx nx C

c. 5 2sin 2 cos 210

xnx nx C

d. 5 2sin 2 cos 210

xnx nx C

3.

23 2

2

2

1

x xdx

x

a. 4 6 2

23 11

4 6 2 2

x x xn x C

b. 4 6 2

23 11

4 6 2 2

x x xn x C

c. 4 6 2

23 11

4 6 2 2

x x xn x C

d. 4 6 2

23 11

4 6 2 2

x x xn x C

4. sin 1 .cos 2

dx

x x is equal to

a.

sin 1cos1.

sec 1

xn C

x


7

b.

sin 1sec1.

sec 1

xn C

x

c. cos1. sin 1 .sec 2n x x

d. sec1. sin 1 .sec 2n x x C

5. 2nx dx is equal to

a. 2

2 2x nx nx C b. 22 2x nx nx C c. 2

2 2x nx nx C

d. 2

2x nx n C

6. 3 1tanx d x

is equal to

a. 2

211

2 2

xn x C b.

221

12 2

xn x C c.

221

12 2

xn x C

d. 2

211

2 2

xn x C

7. 1tan 2 11 . cot

x

e x x d x is equal to

a. 1tan x

e C

b.

1tan x Ce c.

1tan.x

x e C d.

tan 1. xx e C

8. sin 2 tanxd x is equal to

a. 2 cosn x C b. cosn x C c. 2 secn x C d. secn x C

9. bx axa b dx is equal to

a.

bx axa bC

na nb

b. .

bx axa bC

b na nb

c.

bx axa bC

a b

d. None of these

10. 1

2

2sin

1

xdx

x

is equal to

a. 1 1tan sec tann x n x C b. 1 1tan sec tann x n x C


8

c. 1 1tan cos tann x n x C

d. None of these

11. n nx

dxx nx

is equal to

a. 2

n nx C b. 1

2n nx C c.

21

2n n C d. None of these

12. xx n ex dx is equal to

a. xx C b. .x nx C c.

xnX C d. None of these

13. 1 sin

1 cos

x xe dx

x

is equal to

a. cot2

x xe C b. tan

2

x xe C c. sin

2

x xe C d. cos

2

x xe C

14. If 4 3tan tan tan ,xdx a x b x cx d then

a. 1

, 1, 13

a b c b. 1

, 1, 13

a b c c. 1

, 1, 13

a b c

d. 1

, 1, 13

a b c

15. sin 2 . cosx n x dx is equal to

a. 21cos 1 2 cos

2x n x C

b. 21cos 1 2 cos

2n x C

c. 21cos 1 cos

2x n x C

d. None of these

16. 1

.x

x n dxx

is equal to

a. 21 1

12

xx n x x n C

x

b.

21 11

2

xx n x x n C

x


9

c. 21 11

2

xx n x x n C

x

d. None of these

17. .

1

x

x

x edx

e

a. 1 1

2 1 11 1

xx

x

en x e C

e

b.

1 12 1 1

1 1

xx

x

ex e n C

e

c. 1 1

2 1 11 1

xx

x

en x e C

e

d. None of these

18.

2 2

2

2

cos.cos

1

x xec dx

x

is equal to

a. 1cot tanx x C b. 1cot tanx x C c. 1cot tanx x C

d. 1tan cotx x C

19.

3

21

x dx

x is equal to

a. 2 211 2

3x x C b. 2 21

1 13

x x C c. 3/ 2

211

3x C

d. 2 211 2

3x x C

20. sin .cos .cos2 .cos4 .cos8x x x x x dx is equal to

a. 1

cos11696

x C b. 1

cos16256

x C c. 1

cos1616

x C

d. None of these

21. tan .tan 2 .tan3x x x dx is equal to

a. 1 1

sec3 sec2 sec3 2

n x n x n x C

b. 1

sec3

n x

c. tan 2, tan1

d. None of these


10

22.

1

1 1

sin

sin cosdx

x x

is equal to

a. 1 1 24sin cos 1

xx x x C

b. 1 1 22sin cos 1

xx x x C

c. 1 1 22 4sin cos 1

xx x x C

d. None of these

23. 1 2 tanx tanx+secx dx is equal to

a. tan tan tann x x x C b. tan sec tann x x x C

c. sec sec tann x x x C d. sec sec tann x x x C

24.

1 2

2

1 ,

1 1

n n

x

n n

n x xe dx

x x

is equal to

a. 1

1

n

x

n

xe C

x

b.

1

1

n

x

n

xe C

x

c. 1

1

n

x

n

xe C

x

d.

1

1

n

x

n

xe C

x

25. 3

4sin .cos .cos2 2

x xx dx is equal to

a. 1 1

cos cos 2 cos32 3

x x x C b. 1 1

cos cos 2 cos32 3

x x x C

c. 1 1

cos .cos 2 cos32 3

x x x C d. 1 1

cos cos 2 cos32 3

x x x C

Exercise-3

1. If 3

3 5cot tan ,

sin .cos

dxa x b x c

x x where ‘c’ is the constant of integration, then

a. a=2, b=2

3

b. a=2, b=2

3


11

c. a=-2,b=2

3

d. a=-2, b=2

3

2.

4

2

sin

cos

xdx

x is equal to

a. 3 1

sin 2 2 tan2 4

x x C

b. 3 1

sin 2 tan2 4

xx x C

c. 3 1

sin 2 tan2 4

xx x C

d. 3 1

sin 2 tan2 4

xx x C

3. , ; ,bx axa b dx a b R is equal to

a.

.

.

b a

b a

a bC

n a b b.

.

.

bx axa bC

na nb c.

.

.

bx ax

b a

a bC

na nb d. None of these

4. tan

,sin .cos

n xdx

x x is equal to

a. 1

tan2

n x C b. 21tan

2n x C c.

21tan

2n x C d. None of these

5. 2

dx

x x is equal to

a. sin-1 (1-2x)+C b. sin-1 (2x-1)+C c. 1sin 1 2x C d. None of these

6. If 3

2sin sin 2,

x xf x dx

x

0,x then '

0Lim is equal to x

f x

a. 0 b. 1 c. 2 d. 1/ 2

7. If 1tan tan ,

5 4cos 2

dx xa b C

x

then

a. a=2 1

,3 3

b


12

b. a=2 1

,3 3

b

c. 2 1

,3 3

a b

d. 2 1

,3 3

a b

8. tan sec sin ,xe x x dx is equal to

a. tan sin sece x x b. tan .sinxe x C c. tan .cosxe x C d. tan .secxe C

9.

4/5

2 51

dx

x x is equal to

a.

1/551

5

xC

x

b.

1/5

51

5

xC

x

c.

1/5

51 xC

x

d.

1/5

51 xC

x

10. If

4

22

1,

11

x bdx a x C

xx x

then

a. a=1 b. 1, 1a b c. 1, 1a b d. a=1, b=1

11. 3 5

4 54 5

x xx x

x x

e edx ax b n e e C

e e

, then

a. 1 7

,8 8

a b

b. 1 7

,8 8

a b

c. 1 7

,8 8

a b

d. 1 7

,8 8

a b

12. 2

1

1

1

xn

xdx

x

is equal to

a.

21 1

2 1

xn C

x

b.

2

1 1

2 1

xn C

x


13

c.

2

1 1

4 1

xn C

x

d. 1 1

4 1

xn C

x

13. cot cos cosxe x ecx dx is equal to

a. cot .cosxe x C b. cot .sinxe x C c. cot .sinxe x C d. cot .sinxe x C

14. cos 4 1

cot tan

xdx

x x

is equal to

a. 21 1sec2 cos 2

2 4n x x C

b. 21 1sec2 cos

2 4n x x C

c. 21 1cos 2 cos 2

2 4n x x C

d. 21 1cos 2 cos

2 4n x x C

15. If 1 2 1.cot 1. cotx x x x xe e dx a n e be e cx d then

a. 1

, 1, 12

a b c

b. 1

, 1, 12

a b c

c. 1

, 1, 12

a b c

d. 1

, 1, 12

a b c

16. If

77

7

11 ,

1

xdx a n x b n x c

x x

then

a. a=1, b=2

7

b. 2

1,7

a b

c. 2

1,7

a b

d 2

1,7

a b


14

17. If 2 2 2 2. ,x xx e dx e ax bx c d then

a. 1 1 1

, ,2 2 4

a b c

b. 1 1 1

, ,2 2 4

a b c

c. 1

1, 1,2

a b c

d. 1

1, 1,2

a b c

18. If

2

2 2

2

1. 1 1 ,

1

n x xx dx a x n x x bx c

x

then

a. 1, 1a b b. 1, 1a b c. 1, 1a b d. 1, 1a b

19. If 3cot 3 cot 3 tan

, thentan 3tan 3 3 tan

x x xdx ax b n c

x x x

a. a=1,b= 3 b. a=1, b=- 3 c. a=1,b=-1 3 d. a=1 5

,2 4 3

b

20. If 1 2

4 4

sin 2cot tan ,

sin cos

xdx a b x c

x x

then

a. a=1,b=-1 b. a=-1,b=1 c. a=-1,b=-1 d. None of these

21. If

3/ 22 22

6 5

4 64 a x bxxdx C

x x

then

a. a=40

, 13

b b. 1

, 1120

a b c . 40

, 13

a b d. 1

, 1120

a b

22. If 2

3tan tan ,

cos sin 2

dxa x b x c

x x then

a. 2 1

,5 5

a b b. 2

, 55

a b c. 2 1

,5 5

a b d. 2

, 35

a b

23. Let f’ (x)=3x2.sin 1 1 1

cos , 0, 0 0, 0,x x f fx x

then which of the following is not

correct?

a. is continuous at x=0f x b. is non-differentiable at x=0f x

c. is discontinuous at x=0f x d. is differentiable at x=0f x


15

24. If

3

2 3 3 34 3

1,

11

dx x b ca n d

x x xx x

then

a. a=1 1 1

, ,3 3 3

b c b. 2 1 1

, ,3 3 3

a b c c. 2 1 1

, ,3 3 3

a b c

d. 4

25. If 3

3/ 22 2

21 1 ,

1

x dxa x b x c

x

then

a. 1

, 13

a b b. 1

, 13

a b c. 1

, 13

a b

d. 1

, 13

a b


16

INDEFINITE INTEGRATION By Prof. Rajnish Mishra · INDEFINITE INTEGRATION By Prof. Rajnish Mishra 2...

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Transcript of INDEFINITE INTEGRATION By Prof. Rajnish Mishra · INDEFINITE INTEGRATION By Prof. Rajnish Mishra 2...