Indefinite Integration FOR IIT-JEE

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L E V E L - 1 (Objective)

1. If f(x) = 2sinx ₃sin2x

x −

∫

where x ≠ 0 then Limit

x→0

f′(x) has the value ;

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

2. If 1 2 +

∫

sinx dx = A sin x 4 − 4   π then value of A is : (A) 2 2 (B) 2 (C) 1 2 (D) 4 2 3. If y =

(

1+dxx2

)

3 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. If cos cot tan 4x 1 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. dx

x 5+4

∫

_cos = I tan-1 _m _tan x 2    + C then : (A) I = 2/3 (B) m = 3 (C) I = 1/3 (D) m = 2/3 6. Given (a > 0) , 1 xlog_ax

∫

dx = log_ea log_e(log_ex) is true for :

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

7.

( )

cot−

∫

1 _e e x x dx is equal to : (A) 1 2 ln (e 2x_{+ 1) -}cot

( )

−1 e e x x + x + c (B) 1 2 ln (e 2x_{+ 1) +}cot

( )

−1 _e e x x + x + c (C) 1 2 ln (e 2x_{+ 1) -}cot

(

)

−1 _e e x x - x + c (D) 1 2 ln (e 2x_{+ 1) +}cot

( )

−1 _e e x x - x + c 8. tan cot tan cot − − − − − +

∫

11 11 x x x x dx is equal to : (A) 4 π x tan-1 x + 2 π ln (1 + x2) - x + c (B) 4 π x tan-1 x - 2 π ln (1 + x2) + x + c (C) 4 π x tan-1 x + 2 π ln (1 + x2) + x + c (D) 4 π x tan-1 x - 2 π ln (1 + x2) - x + c

INDEFINITE

INTEGRATION

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9. If

( )

x x x 4 2 2 1 1 + +

∫

dx = A ln |x|+ B x

1+ 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.

∫

(

2 tan x

)

+ c (B) x - 1 2 arctan tan x 2    _ + c

(C) x - 2 arctan

(

2 tan x

)

+ c (D) x - 2 arctan tan 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 13. x x x 2 2 2 1 + +

∫

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 l tan−1 - cot x + c

where 'c' is constant of integration .

14. 3 5 4 5 e e e e x x x x + − − −

∫

dx = Ax + B ln | 4e2x_{- 5 | + 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 − +

∫

1₁. 12 dx equals : (A) sin -1 1 x + x x 2 1 − (B) x x 2 1 − + cos -1 1 x + c (C) sec -1 _x_- x x 2 1 − + c (D) tan -1 _x2+₁_- x x 2 1 − + c 16. dx x−x

∫

2 equals : (A) 2 sin -1

x + c (B) sin -1 _{(2x - 1) + c (C) c - 2 cos} -1_{(2x - 1) (D) cos} -1₂

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17.

∫

2mx_{. 3}nx_{dx when m, n}_∈_{N is equal to :} (A) 2 3 2 3 mx nx m n n n + + l l + c (B) ( ) e m n n n m nl n nl x l l 2 3 2 3 + + + c (C)

(

)

2 3 2 3 mx nx m n n . . l + c (D)

( )

m n m n n n x x .2 .3 2 3 l + l + c 18. dx x x cos3 . sin2

∫

equals : (A) 2 5 (tan x) 5/2_{+ 2} tan x + c (B) 2 5 (tan 2 _{x + 5)} tan x + c (C) 2 5 (tan 2 _{x + 5)} 2 tan x + c (D) none 19. If dx x x sin3 cos5

∫

= a cot x + b tan3x + c where c is an arbitrary constant of integration then the values of ‘a’ and ‘b’ are respectively :

(A) -2 & 2 3 (B) 2 & -2 3 (C) 2 & 2 3 (D) none 20. 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

21. cos cos sin sin 3 5 2 4 x x x x + +

∫

dx :

(A) sinx - 6 tan-1 _(sin_{x) + c} _{(B) sin x - 2 sin}-1 _{x + c}

(C) sinx - 2 (sinx)-1_{- 6 tan}-1 _{(sin x) + c} _{(D) sin}_{x - 2 (sin}_x)-1_{+ 5 tan}-1 _(sin_{x) + c}

22. ln x x x (tan ) sin cos

∫

dx equal : (A) 1 2 ln 2_{(cot x) + c} _(B)1 2 ln 2_{(sec x) + c} (C) 1 2 ln 2_(sin_x_sec_{x) + c} _(D)1 2 ln 2_{(cos x cosec}_{x) + c} 23.

∫

sec2 ₂ 4 x−   π dx equals : (A) c - 1 2 cot 2x+ 4    π (B) 1 2 tan 2x− 4    π + c (C) 1

2(tan 4x - sec 4x) + c (D) none

24. Primitive of

(

3 11

)

4 4 2 x x x − + + w.r.t. x is : (A) x x4 + +x 1 + c (B) -x x4 + +x 1 + c (C) x x x + + + 1 1 4 + c (D) -x x x + + + 1 1 4 + c

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1.

∫

cos cos cos 5 4 1 2 3 x x x dx + − 2.

∫

cos x . e x_{. x}2_dx _3.

∫

sin ( ) sin ( ) x a x a − + dx 4.

∫

cot ( sin ) (sec ) x dx x x 1− +1 5.

∫

cos cot cos cot . sec sec ec x x ec x x x x − + 1+ 2 dx 6.

∫

d x x x sin +sec

7.

∫

tan x . tan 2x . tan 3x dx 8.

(

)

dx x x sin sin 2 +

∫

_α 9.

∫

x x x x 2 2 ( sin + cos ) dx 10.

∫

ln

(

)

x x x cos cos sin + 2 2 dx 11. sin sin cos x x + x

∫

dx 12.

∫

e x x x x x

sin _. cos sin

cos 3 2 − _dx 13.

∫

₍ d x ₎ a+ bcosx 2 (a > b) 14.

∫

cos sin 2 x x dx 15. cot tan sin x x x − +

∫

₁ ₃ ₂ dx 16.

(

5 41

)

4 5 5 2 x x x x + + +

∫

dx 17.

(

x4dx−1

)

2

∫

18.

∫

ex

(

x

)

x 2 2 1 1 + + ( ) dx 19.

∫

x+ x2+ 2 dx 20.

∫

x

[

l n x

(

)

x

]

x 2 2 4 1 1 2 + + −         ln dx 21.

∫

l n x x (ln ) (ln ) +       1 2 dx 22.

∫

( ) ( )

[

x−13d xx+2 5

]

1 4/ 23.

(

)

dx x3+3x2 +3x +1 x2 +2x−3

∫

24.

∫

( ) ( ) ax b dx x c x ax b 2 2 2 2 2 − − + 25.

∫

e

(

x

)

x x x 2 1 1 2 2 − − − ( ) dx 26.

∫

(

)

x x x 7 −10− 2 3 2/ dx 27.

∫

(

)

x x x ln / 2 3 2 1 − dx 28.

∫

1 1 3 − + x x dx x 29. 2 3 2 3 1 1 − + + −

∫

x_x x_x dx 30.

∫

₍

x

₎

x x d x x + + + + 2 3 3 1 2 31. dx x3 (1+x)3

∫

32.

∫

2 2 2 − −x x x dx 33.

∫

d x x x x ( −α) ( −α) ( −β) 34. Integrate 1 2 f′(x) w.r.t. x 4 _{, where f (x) = tan} -1_{x + ln} ₁₊_x_{- ln} ₁₋_x

L E V E L - 2 (Subjective)

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LEVEL - 3

(Questions asked from previous Engineering Exams)

1. Find the indefinite integral

∫

_{1 3} 1 _{1 4}

[

1

]

1 6 1 3 1 2 ( ) ( ) ( ) ( ) ( ) / / / / / x x l n x x x + + + +         dx 2. Evaluate

∫

3 1 1 3 1 x x x + − + ( ) . ( ) dx . 3. Evaluate

∫

f x x ( ) 3 1

− dx ; where f(x) is a polynomial of second degree in x such that

f(0) = f(1) = 3 f(2) = - 3 .

4. Evaluate ,

∫

cos 2θ . ln cos sin cos sin

θ θ

+

− dθ .

5. Evaluate

∫

cos . sin cos . ( cos ) 2 4 1 2 4 2 x x x + x dx . 6. Integrate ,

(

xx

)

xx 3 2 2 3 2 1 1 + + + +

∫

( ) dx .

7. Let f(x) =

∫

ex_{(x - 1) (x - 2)}_d_{x then}_f_{decreases in the interval :}

(A) (-∞, 2) (B) (-2, -1) (C) (1, 2) (D) (2, ∞) 8. Evaluate ,

∫

sin -1 2 2 4 2 8 13 x x x + + +         dx .

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1. b 2. d 3. d 4. b 5. a,b 6. b 7. c 8. d 9. c 10. a 11. a 12. b

LEVEL - 1 (Objective questions)

ANSWER KEY

13. b,c,d 14. d 15. c 16. a,b,d 17. b,c 18. b 19. a 20. c 21. c 22. a,c,d 23. a,b,c 24. b

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1. −(sinx+sin2x) 2 + c 2. 1 2 e x

[

(

_x2

)

_x _x 2 _x

]

1 1 − cos +( − ) . sin _{+ c}

3. cos a . arc cos cos

cos x a     

 - sin a . ln

(

sinx + sin2x−sin2a

)

_{+ c}

4. 1 2 ln tan x 2 + 1 4 sec² x 2 + tan x 2 + c 5. sin-1 1 2 2 2 sec x    + c 6. 1 2 3 3 3 l n x x x x arc x x c + − − + + + + sin cos

sin cos tan (sin cos )

7. − − +



 

ln(sec )x 1 ln(sec x) ln(sec x)

2 2

1

3 3 + c

8. - 1

sinα ln

[

cotx+cotα+ cot x+ cot cotα x−

]

2 2 1 + c 9. sin cos sin cos x x x x x x c − + + 10. cos sin 2x

x - x - cotx . ln

(

e

(

cosx+ cos2x

)

+ c

11. ln(1 + t) - 1 4 ln(1 + t 4_{) +} 1 2 2 ln t t t t 2 2 2 1 2 1 − + + + - 1 2 tan -1 _t2_{+ c where t =} _{cot x} 12. esinx _{(x - secx) + c} 13. −

(

₋

)

₊ +

(

₋

)

− + b x a b a b x a a b arc a b a b x sin

( cos ) / tan . tan

2 2 2 2 3 2 2 2+ c 14. - 1

(

)

2 2 2 1 2 1 1 2 2 2 ln x x l x x n x x cot cot cot cot cot cot − − + − + − +         + c LEVEL - 2 (Subjective Questions)

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15. tan-1 2sin2 sin cos x x+ x    _ + c 16. - x x x + + + 1 1 5 + c 17. 3 8 tan -1 _{x -}

(

xx

)

4 4 −1 - 3 16 ln x x − +    1_ 1 + c 18. e x x x − + 1 1+ c 19. 1 3

(

x + x +

)

2 3 2 2 / -

(

)

2 2 2 1 2 x+ x + / _{+ c} _20.

(

x

)

x x x 2 2 3 2 1 1 9 2 3 1 1 + + − _ + _     . ln 21. xln (lnx) - x l n x + c 22. 4 3 1 2 1 4 x x c − +       + / 23. x x x 2 2 2 3 8 1 + − + ( ) + 1 16 . cos -1 2 1 x+    _ + c 24. sin−  +      + 1 2 ax b cx k 25. ex 1 1 + − x x + c 26. 2 7 20 9 7 10 2 ( x ) x x c − − − + 27 arc x ln x x c sec − − + 2 1 28. ln u u u u c where u x x | | tan 2 4 2 1 2 3 1 1 3 1 2 3 1 1 − + + + + ₊ ₌ − + − 29. 8

(

)

3 1 2 5 5 1 5 1 1 1 1 2 tan− + − sin− +               − − − t n t t x x l + c where t = 1 1 + − x x 30. 2 3 arc 3 1 x x c tan ( + ) + 31. 15 5 2 4 1 2 2 x x x x + − + + 15 8 ln 1 1 1 1 + − + + x x + c 32. − − − + − + − −         − −  +  2 2 4 4 2 2 2 2 1 3 2 2 1 x x x n x x x x x l sin + c 33. _{α β}−₋2 . x₋−β_α + x c 34. - ln(1- x4₎₊_c

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1. I = I₁ + I₂ + c , where ; I1 y y y y y y y y n y 8 7 6 5 4 3 2 12 8 8 7 28 6 56 5 70 4 56 3 28 2 8 1 =  − + − + − + − +      ; where y = x1/12_{+ 1}

( )

I2 ez z e z z ez z z c 3 2 2 2 2 1 3 9 1 2 18 1 3 = _ − _{ −} _ − _{ +} − − + ; where z = ln (1 + x1/6₎ 2. 1 4 1 1 1 2 1 2 l nx x x x c + − − + − + ( ) 3. ln x x x 2 1 1 + + −      + 2₃ arctan_2x₃+1_ + c 4. (b) 1 2 (sin2θ) ln cos sin cos sin θ θ θ θ + −      - 1 2 _{ln (sec}₂θ_{) + c} 5. 2 ln (1+cos 2x) + 2 1+cos x2 - ln (1 + cos 2 _{2x) +}_c 6. 3 2 tan -1 _{x -} 1 2 ln(1 + x) + 1 4 ln (1+x 2_{) +} x x 1+ 2 + c 7. C 8. -3 2 2 2 3 2 2 3 1 2 4 8 13 1 2 x x x x + + ₋ ₊ ₊     − tan log

(

)

LEVEL - 3 (Questions asked from previous Engineering Exams)