Indefinite integration.pdf

8/21/2019 Indefinite integration.pdf

1/12

INDEFINITE INTG. # 1

IIT-ians PACE ;ANDHERI / DADAR / CHEMBUR / THANE ; Tel : 26245223 / 09 ; .www.iitianspace.com (1)

D EF IN I T IONS ND RESULTS

1. If f & g are functions of x such that g(x) = f(x) then ,

f(x)dx = g(x)+ c ddx

{g(x)+c} = f(x), where c is called the constant of integration.

2. STANDARD RESULTS :

(i) (ax + b)ndx = ax b

a n

n

1

1+ c n 1 (ii) dx

ax b=

1

aln (ax + b) + c

(iii) eax+bdx = 1a eax+b + c (iv) apx+q dx = 1p

a

na

px q

(a > 0) + c

(v) sin (ax+ b) dx = 1a

cos (ax+ b)+ c (vi) cos (ax + b) dx = 1a

sin (ax+ b) + c

(vii) tan(ax+ b) dx = 1a

ln sec(ax + b)+ c (viii) cot(ax+b)dx = 1a

ln sin(ax +b)+ c

(ix) sec (ax + b) dx = 1a

tan(ax + b) + c

(x) cosec(ax + b) dx = 1a

cot(ax + b)+ c

(xi) sec (ax + b) . tan (ax + b) dx = 1a

sec (ax + b) + c

(xii) cosec (ax + b) . cot (ax + b) dx = 1a

cosec (ax + b) + c

(xiii) secx dx = ln (secx + tanx) + c OR ln tan 4 2

x

+ c

(xiv) cosec x dx = ln (cosecx cotx) + c OR ln tan x2

+ c OR ln (cosecx + cotx)

(xv) d xa x2 2 = sin1 x

a+ c (xvi) d xa x2 2 =

1a

tan1 xa

+ c

(xvii) d xx x a

2 2=

1

asec1

x

a+ c

(xviii) d xx a2 2

= ln x x a 2 2 OR sinh1 xa

+ c

(xix) d xx a2 2

= ln x x a 2 2 OR cosh1 xa

+ c

IIT ians P A C E216 - 217, 2nd floor, Shoppers point, S. V. Road. Andheri (West) Mumbai 400 058 . Tel: 2624 5223 / 09


2/12



(xx) d xa x2 2

=1

2aln

a x

a x

+ c (xxi) d xx a2 2

=1

2aln

x a

x a

+ c

(xxii)

a x2 2

dx =

x

2

a x

2 2

+

a 2

2sin1

x

a+ c

(xxiii) x a2 2 dx = x2

x a2 2 +a2

2sinh1

x

a+ c

(xxiv) x a2 2 dx = x2

x a2 2 a 2

2cosh1

x

a+ c

(xxv) eax. sin bx dx = ea b

ax

2 2(a sin bx b cos bx) + c

(xxvi) eax . cos bx dx = ea b

ax

2 2

(a cos bx + b sin bx) + c

3. INTEGRALS OF THE TYPE :

(i) [ f(x)]nf(x) dx OR

f xf x

n

( )

( ) dx put f(x) = t & proceed .

(ii)dx

ax bx c2 ,

dx

ax bx c2 , ax bx c2 dx

Express ax2+ bx + c in the form of perfect square & then apply the standard results .

(iii)px q

ax bx c

2 dx ,px q

ax bx c

2

dx .

Express px + q = A (differential coefficient of denominator) + B .

(iv) ex [f(x) + f(x)] dx = ex . f(x) + c (v) [f(x) + xf(x)] dx = x f(x) + c

(vi) d xx xn( )1

n N Take xn common & put 1 + x n= t .

(vii)

dx

x xnn

n21

1 ( ) n N , take xn common & put 1+xn= tn

(viii)

dx

x xn n

n

1

1

/ take xncommon as x and put 1 + xn= t .

(ix) d x

a b x sin2 OR

d x

a b x cos2OR

d x

a x b x x c xsin sin cos cos2 2

Multiply Nr. . & D

r. . by sec x & put tan x = t .

(x) d x

a b x sinOR

d x

a b x cos OR

d x

a b x c x sin cos Hint :

Convert sines & cosines into their respective tangents of half the angles , put tanx

2= t


3/12



(xi) a x b x cx m x n

.cos .sin

.cos .sin

dx . Express Nr A(Dr) + Bd

d x(Dr) + c & proceed .

(xii)

x

x K x

2

4 2

1

1

dxOR

x

x K x

2

4 2

1

1

dx where K is any constant .Divide Nr & Dr by x & proceed .

(xiii)dx

ax b px q( ) & dx

ax bx c px q2 ; put px + q = t2.

(xiv) dx

ax b px qx r ( )

2

, put ax + b =1

t ;

dx

ax bx c px qx r 2 2

, put x =1

t

(xv)x

x

dx or x x ; put x = cos2 + sin2

x

x

dx or x x ; put x = sec2 tan2

dx

x x

; put x = t2 or x = t2.

EXERCISE I

1. If f(x) =2 2

3

sin sinx x

x

dx where x 0 then Limitx 0 f(x) has the value ;

(A) 0 (B) 1 (C) 2 (D) not defined

2. If 12

sinx

dx = A sin x

4 4

then value of A is :

(A) 2 2 (B) 2 (C)1

2(D) 4 2

3. If y =

dx

x1 23 2

/ and y = 0 when x = 0, then value of y when x = 1 is :

(A)2

3(B) 2 (C) 3 2 (D)

1

2

4. Ifcos

cot tan

4 1x

x x

dx = A cos 4x + B where A & B are constants, then :

(A) A = 1/4 & B may have any value (B) A = 1/8 & B may have any value(C) A = 1/2 & B = 1/4 (D) none of these

5. cot x sec 4 x d x =

(A) 2 tan x +2

5 tan5 x + c (B) 2 tan x +

2

5 tan5 x + c

(C) tan x +2

5 tan5 x + c (D) tan x +1

5 tan5 x + c


4/12



6. Given (a > 0) ,1

x xalog dx = loge a loge (loge x) is true for :

(A) x > 1 (B) x > e (C) all x R (D) no real x .

7. cot1 e

e

x

xdx is equal to :

(A)1

2ln (e2x+ 1)

cot1 ee

x

x+ x + c (B)

1

2ln (e 2x+ 1) +

cot1 ee

x

x+ x + c

(C)1

2ln (e 2x+ 1)

cot1 ee

x

xx + c (D)

1

2ln (e2x+ 1) +

cot1 ee

x

xx + c

8.tan cot

tan cot

1 1

1 1

x x

x x

dx is equal to :

(A)4

x tan1 x +

2

ln (1 + x2) x + c (B)

4

x tan1 x

2

ln (1 + x2) + x + c

(C)4

x tan1 x +

2

ln (1 + x2) + x + c (D)

4

x tan1 x

2

ln (1 + x2) x + c

9. If

x

x x

4

22

1

1

dx = A ln x+

B

x1 2+ c , where c is the constant of integration then

(A) A = 1 ; B = 1 (B) A = 1 ; B = 1 (C) A = 1 ; B = 1 (D) A = 1 ; B = 1

10.

n x

x n x

| |

| |1dx equals :

(A)2

31 n x (lnx 2) + c (B)

2

31 n x (lnx+ 2) + c

(C)1

31 n x (lnx 2) + c (D) 2 1 n x (3 lnx 2) + c

11. Antiderivative ofsin

sin

2

21

x

x w.r.t. x is :

(A) x 22arctan 2 tanx + c (B) x 12 arctan

tan x2

+ c

(C) x 2 arctan 2 tanx + c (D) x 2 arctantan x

2

+ c

12. sin x . cos x . cos 2x . cos 4x . cos 8x . cos 16 x dx equals :

(A)sin 16

1024

x+ c (B)

cos 32

1024

x+ c (C)

cos 32

1096

x+ c (D)

cos 32

1096

x+ c


5/12



13. 1

1

x

xdx =

(A) x 1 x 2 1 x + cos1 x + c (B) x 1 x + 2 1 x + cos1 x + c

(C) x 1 x 2 1 x cos 1 x + c (D) x 1 x + 2 1 x cos 1 x + c

14.3 5

4 5

e e

e e

x x

x x

dx = Ax + B ln 4 e2x5+ c then :

(A) A = 1, B = 7/8; C = const. of integration(B) A = 1, B = 7/8; C = const. of integration(C) A = 1/8, B = 7/8 ; C = const. of integration(D) A = 1, B = 7/8; C = const. of integration

15.x

x x

1

1

12

. dx equals :

(A) sin11

x+

x

x

2 1(B)

x

x

2 1+ cos1

1

x+ c

(C) sec1 x xx

2 1+ c (D) tan1 x2 1

x

x

2 1+ c

16. tan 32x sec 2x dx =

(A)1

3sec32

x

1

2sec 2

x + c (B)

1

6sec32

x

1

2sec 2

x + c

(C) 16 sec32x 12 sec 2x + c (D) 13 sec

32x + 12 sec 2x + c

17. ( )x

x xex

1

12

dx =

(A) ln

x e

x e

x

x1

+

1

1 ex+ c (B) l

n

x e

e

x

x1

+

1

1 x ex+ c

(C) ln

x e

x e

x

x1

+

x

x ex1+ c (D) l

n

x e

x e

x

x1

+

1

1 x ex+ c

18.dx

x xcos . sin3 2 equals :

(A)2

5(tan x)5/2+ 2 tanx + c (B)

2

5(tan2 x + 5) tanx + c

(C)2

5(tan2 x + 5) 2tanx + c (D) none


6/12



19. Ifdx

x xsin cos3 5 = a cot x + b tan3 x + c where c is an arbitrary constant of

integration then the values of a and b are respectively :

(A) 2 & 23 (B) 2 & 23

(C) 2 & 23

(D) none

20. cos

sin sin

3

2

x

x x d

x =

(A) ln sin x+ sin x + c (B) l

n sin xsin x + c

(C) ln sin xsin x + c (D) l

n sin x+ sin x + c

21. cos cos

sin sin

3 5

2 4

x x

x x

dx :

(A) sinx 6 tan1 (sin x) + c (B) sin x 2 sin1 x + c(C) sinx 2 (sinx)16 tan1 (sin x) + c (D) sinx 2 (sinx)1+ 5 tan1 (sin x) + c

22. 1

6 6cos sinx x d

x =

(A) tan1(tan x + cot x) + c (B) tan1(tan x + cot x) + c(C) tan1(tan x cot x) + c (D) tan1(tan x cot x) + c

23. Primitive of

3 1

1

4

42

x

x x

w.r.t. x is :

(A)x

x x4 1

+ c (B) x

x x4 1

+ c (C)x

x x

1

14+ c (D)

x

x x

1

14+ c

24.dx

x5 4 cos = tan1 m

xtan

2

+ C then :

(A) = 2/3 (B) m = 1/3 (C) = 1/3 (D) m = 2/3

25.x x

x

2 2

21

cos

cosec2 x dx is equal to :

(A) cot x cot 1 x + c (B) c cot x + cot 1 x

(C) tan 1 x cos

sec

ec x

x+ c (D) e n x tan

1cot x + c

where 'c' is constant of integration .

26.dx

x x 2 equals :

(A) 2 sin 1 x + c (B) sin1 (2x 1) + c

(C) c 2 cos1(2x 1) (D) cos12 x x 2 + c


7/12



27. 2mx. 3nx dx when m, n N is equal to :

(A)2 3

2 3

mx nx

m n n n

+ c (B)

e

m n n n

m n n n x

2 3

2 3

+ c

(C)

2 3

2 3

mx nx

m nn

.

.+ c (D)

mnm n n n

x x. .2 3

2 3 + c

28. If eu. sin 2x dx can be found in terms of known functions of x then u can be :(A) x (B) sin x (C) cos x (D) cos 2x

29. sec2 24

x

dx equals :

(A) c 1

2cot 2

4x

(B)1

2tan 2

4x

+ c (C)1

2(tan 4x sec 4x) + c

(D) none

30. n x

x x

(tan )

sin cos dx equal :

(A)1

2ln2(cot x) + c (B)

1

2ln2(sec x) + c

(C)1

2ln2(sinx secx) + c (D)

1

2ln2(cos x cosec x) + c

EXERCISE II

LEVEL I

Evaluate the following :

1. cos cos

cos cos

2 2x

x

dx 2. sin cos

sin cos

6 6

2 2

x x

x x

dx

3. x

a bx

2

2( )dx 4. sin 4x cos4x dx

5.

cos 2x cos 4x cos 6x dx

6.

1

sin ( ) cos ( )x a x b dx

7. tan

tan

x

a b x 2dx 8. 1 2

1 2 tan (tan sec ) /x x x dx

9. 1

3cos cosx xdx 10.

e x

x

sin

1 2

21dx

11. x x

x

1

2dx 12. 5

1x x tan .x

x

2

2

2

1

dx


8/12



13. x

a x3 3dx 14. cosec x 1 dx

15. 2 3

3 182x

x x

dx 16. sin

sin

x

x3 dx 17. x x

x

sin

cos1 dx

18. x x

x

2 1

23 2

1

sin/

dx 19. ex

x

x( ) 1 2dx 20.

( )x

x x

1

1

2

4 2dx

LEVEL II

Evaluate the following :

21. 1

3sin sin ( )x x dx , n

, n Z

22.

1

12 43 4

x x /

dx 23. 2 2

6 42sin cos

cos sin

dx

24. x a xa x

2 2

2 2

dx 25. sin 1

x

a x dx

26. tan tan

tan

3

31d 27.

sin

sin

x

x4dx

28. 1 2sin sinx x dx 29. 11 12 2( ) ( )x x dx

30. tan cot d 31. ex

x x

x

3

22

2

1

dx

32. sin cos

sin

x x

x

9 16 2

dx 33. 3 4 23 2

sin cos

sin cos

x x

x xdx

34. cos 2x ln (1 + tan x) dx 35. d xx xsin tan

36. 1

1 4 sin xdx 37. cos

sin

3

11

x

xdx

38. x

x x 3dx 39.

1

cos cos

cos cos

x

xdx

40. d x

x x x( ) ( ) ( ) 41. sec4 x cosec2x dx

42. cos sin

( cos sin )

2

2

2

2

x x

x x

dx 43. cos cos

cos

5 4

1 2 3

x x

xdx


9/12



44. cos x . ex. x2dx 45. sin ( )sin ( )

x a

x a

dx

46.

cot

( sin ) (sec )

x dx

x x1 1 47.

cos cot

cos cot

.sec

sec

ec x x

ec x x

x

x

1 2dx

48. d x

x xsin sec 49. tanx. tan 2x. tan 3x dx

50.

dx

x xsin sin 2

51. x

x x x

2

2( sin cos )

dx

52. n x x

x

cos cos

sin

22 dx 53.

sin

sin cos

x

x x dx

54.

e

x x x

x

xsin.

cos sin

cos

3

2

dx 55.

d x

a b x cos2 (a > b)

56. cos

sin

2 x

xdx 57.

cot tan

sin

x x

x

1 3 2 dx

58. 5 4

1

4 5

52

x x

x x

dx 59.

dx

x42

1 60. ex

x

x

2

2

1

1

( )dx

61. x x 2 2 dx 62. x l n x x

x

2 2

4

1 1 2

lndx

63. l n x x(ln ) (ln )

12

dx 64.

d x

x x 1 23 51 4/

65.

dx

x x x x x3 2 23 3 1 2 3 66.

( )

( )

ax b dx

x c x ax b

2

2 2 2 2

67. e x

x x

x 2

1 1

2

2

( )dx 68.

x

x x7 10 23 2

/ dx

69. x x

x

ln/2 3 21

dx 70. 113

x

x

dx

x71.

2 3

2 3

1

1

x

x

x

xdx

72. x

x x

d x

x

2

3 3 12 73.

dx

x x3 31( ) 74. 2

2

2

x x

xdx

75. Integrate1

2f(x) w.r.t. x4 , where f (x) = tan 1x + ln 1x ln 1x


10/12



ANSWER SHEET

EXERCISE I

1. B 2. D 3. D 4. B 5. B 6. A 7. C

8. D 9. C 10. A 11. A 12. B 13. A 14. D

15. C 16. C 17. D 18. B 19. A 20. B 21. C

22. C 23. B 24. AB 25. BCD 26. ABD 27. BC

28. ABCD 29. ABC 30. ACD

EXERCISE II

1. 2 sin x + 2x cos + c 2. tan

x cot

x 3

x + c

3.13b

b x a bx aa

a bx

2

2

log( )

| | + c 4. 1128

3 41

88x x x

sin . sin + c

5.1

4 x

x x x

sin sin sin12

12

8

8

4

4+ c 6.

1

cos ( )a blog

e

sin ( )

cos ( )

x a

x b

+ c

7.1

2 ( )b alog a x b xcos sin2 2 + c 8. log sec x + tan x+ log sec x+ c

9.1

4

[cosec x - log sec x + tan x

]+ c 10.

e xsin

1 2

2+ c

11. (x + 1) + 2 x 1 2 log x + 2 2 tan 1

x 1 + c 12. 15log

51x x

tan + c

13.2

3sin 1

x

a

3 2

3 2

/

/

+ c 14. log sin sin sinx x x

1

22 + c

15. log x x2 3 18 2

3log

x

x

3

6+ c 16.

1

2 3log

3

3

tan

tan

x

x+ c

17. x cot x2 + c18. x x

xsin

1

21 12 sin1

2x + 1

2log 1 2 x + c 19. 1 1x . e

x+ c

20.1

3tan 1

x

x

2 1

3

2

3tan 1

2 1

3

2x

+ c 21.

2

sin sin ( )

sin

x

x

+ c

22. 114

1 4

x

/

+ c 23. 2 log sin sin2 4 5 + 7 tan 1(sin 2) + c

24.1

2a2sin1

x

a

2

2

+

1

2 a x4 4 + c 25. (a + x) arc tan

x

a a x + c


11/12



26. 1

31 + tan +

1

6log tan2 tan + 1+ 1

3tan1

2 1

3

tan

+ c

27. 1

8 log

1

1

sin

sin

x

x +

1

4 2 log1 2

1 2

sin

sin

x

x + c

28.1

6log 1 cos x+

1

2log 1 + cos x+

2

5log 3 + 2 cos x+ c

29.1

2log x + 1

1

2 1( )x 1

4log x2 1 + c 30. 2 tan1

tan

tan

1

2+ c

31. exx

x

1

12+ c 32.

1

40log

5 4

5 4

(sin cos )

(sin cos )

x x

x x+ c 33. 2 3

21x arc

xc

tan tan

34.1

2 [sin 2x ln (1 + tan x) - x + ln (sin x + cos x)]+ c 35. 12 2 14 22l n x x ctan tan

36.1

2 2tan 1 2 tan x + 1

2tan x + c 37.

2

5cot5/2x

2

9cot 9/2x + c

38. x 6

5x5/6+

3

2x2/32 x + 3 x1/36 x1/6+ 6 l

n (1 + x1/6) + c

39.x cos + sin logcos ( )

cos ( )

1212

x

x+ c 40.

2

.x

xc

41.1

3 tan3 x + 2 tan x cot x + c 42.

1

2 tan x 1

5 x 2

5 log2 cos x sin x+ c

43. (sinsin

)xx2

2+ c 44.

1

2ex x x x x2 21 1 cos ( ) . sin + c

45. cos a . arc cos coscos

x

a

sin a . ln sin sin sinx x a 2 2 + c

46.1

2ln tan

x

2+

1

4sec

x

2+ tan

x

2+ c 47. sin 1

1

2 2

2sec

x

+ c

48.1

2 3

3

3l n

x x

x xarc x x c

sin cos

sin costan (sin cos )

49.

n x n x n x(sec ) (sec ) (sec )1

22

1

33 + c

50. 1

sinln cot cot cot cot cotx x x 2 2 1 + c 51. sin cos

sin cos

x x x

x x xc

52.cos

sin

2x

xx cot x . ln e x xcos cos 2 + c

53. ln(1 + t) 1

4ln(1 + t4) +

1

2 2ln

t t

t t

2

2

2 1

2 1

1

2tan1 t2+ c where t = cotx


12/12



54. esinx (x secx) + c 55.

b x

a b a b x

a

a barc

a b

a b

xsin

( cos )tan . tan

/2 2 2 2 3 2

2

2+ c

56. 12 22 1

2 11

2

22 n

x x

x xn x x

cot cot

cot cotcot cot

+ c 57. tan

1 2 2sinsin cos

xx x

+ c

58. x

x x

1

15 + c 59. 3 tan1 x

x

x4 14

3

16ln

x

x

1

1+ c 60. ex

x

x

1

1+ c

61.1

3 x x 2

3 2

2/

2

221 2

x x / + c 62.

x xx x

2 2

3 2

1 1

92 3 1

1

. ln

63. xln (lnx) x

l n x+ c 64. 4

3

1

2

1 4

x

xc

/

65. x x

x

2

2

2 3

8 1

( )

+1

16 . cos1

2

1x

+ c

66. sin

1

2ax b

cxk 67. ex

1

1

x

x+ c 68.

2 7 20

9 7 10 2

( )x

x xc

69. arc x

l n x

xcsec

2 1

70. nu

u u

uc where u

x

x

| |tan

2

4 2

12

31

13

1 2

3

1

1

71. 83

1

2 5

5 1

5 111 1 2tan sin

t nt

tx x + c where t =

1

1

x

x

72.2

3 3 1

arcx

x

ctan

( )

73. 15 5 2

4 1

2

2

x x

x x

+

15

8

ln1 1

1 1

x

x+ c

74.

2 2

4

4 2 2 2 2 1

3

2 21x x

xn

x x x

x

x sin + c 75. ln(1 x4) + c

Indefinite integration.pdf

Documents

Transcript of Indefinite integration.pdf