Definite INTEGRATION ASSIGNMENT FOR IIT-JEE

L E V E L - 1 ( Fundamentals of Definite Integration) 1.

∫e 1

21nx

0

dx =

(a) 0 2.

∫

π /4

(b)

∫

π /2

0

π 4

∫

π /2

0

∫

2

1

π −1 4

(d)

(b) log e 2

(c)

π 2

(d) 0.

(c)

1 (1 − eπ / 2 ) 2

π (d) 2 (e / 2 + 1) .

(b)

1 π /2 (e + 1) 2

(b) e −

e2 2

e2 −e (c) 2

(d) None of these.

cos x

∫ (1 + sin x )(2 + sin x ) dx = 0

(a) log

8.

π . 4

(c)

1 1  e x  − 2  dx = x x 

π /2

7.

1 . 4

e x sin xdx =

e2 +e (a) 2 6.

(d)

x + sin x dx = 1 + cos x

1 π /2 (e − 1) 2

(a) 5.

1 3

π 4

(b) 1 +

(a) − log e 2 4.

(c)

tan 2 xdx =

0

(a) 1 − 3.

1 2

∫

π /2

4 3

(b) log

1 + cos x 5

π /3

(1 − cos x )2 5 (a) 2

∫

2

1

(a)

1 3

(c) log

3 4

(d) None of these.

dx =

(b)

3 2

(c)

1 2

(d)

2 . 5

−1

1 x e dx = x2 e +1

(b)

e −1

(c)

e +1 e

(d)

e −1 e .

∫ sin 1

9.

−1

0

(a)

 2x  dx =  2   1+ x 

π − 2 log 2 2

(b)

∫ (ax 2

10. The value of

3

−2

(a) The value of a 11.

∫

π /4

π /6

∫

b

a

∫ tan 1

0

(a)

dx

2

0

∫

π /2

π /4

∫

17.

(a)

(c) The value of c

(d) The value of a and b.

(b) log 3

(c) log 9

(d) None of these.

b (b) log ( a b ) log   a

(c)

1 b log ( a b ) log   2 a

1 (b) π − log 2 2

(c)

π − log 2 4

b a

(c) a b

(b) 1 − 2

sin −1 x

(1 − x 2 )

3/ 2

(d)

1 a log ( a b ) log   2 b

(d) π − log 2 .

(d)

1 . ab

(c)

2 +1

(d) None of these .

dx =

π 1 + log e 2 4 2

π /2

0

(b) The value of b

(b)

2 −1

0

∫

+ bx + c ) depends on

cos θ cos ec 2θ dθ =

1/ 2

(a)

π + log 2 . 4

=

a b

(a) 16.

(d)

x dx =

∫ (ax + b (1 − x )) (a)

15.

−1

π 1 − log 2 4 2

1

14.

π − log 2 4

log x dx = x

 log b  (a) log    log a  13.

(c)

cos ec 2 x dx =

(a) log 3 12.

π + 2 log 2 2

(b)

π 1 − loge 2 4 2

(c)

π + log e 2 2

(d)

π − loge 2 . 2

dx = 2 + cos x

1  1  tan −1   3  3

(b)

3 tan −1

( 3)

(c)

2  1  tan −1   3  3

(d) 2 3 tan −1

( 3).

tan−1 x dx = 18. ∫0 1 + x2 1

π2 8

(a)

(b)

19. The value of integral (a) 2 20.

∫

2π

0

∫

2/π

1/ π

π2 16

(c)

π2 4

(d)

π2 . 32

sin (1/ x ) dx = x2

(b) -1

(c) 0

(d) 1.

(c) 0

(d) None of these.

x π e x / 2 .sin  +  dx = 2 4

(b) 2 2

(a) 1 e− x dx = 21. ∫0 1 + e− x 1

 1+ e  1

(a) log   − +1  e  e 22.

∫

23.

0

∫

1 log 3 20

π /2

π /4

(b) log 3

24.

∫

25.

x sin −1 x 1 − x2

0

(a)

∫

1 3π + 2 12

π

0

(c)

1 log 5 20

(d) None of these.

(b) −eπ / 4 log 2

(c)

1 π /4 e log 2 2

1 (d) − eπ / 4 log 2 . 2

(b)

1 3π − 2 12

(c)

1 3π − 2 12

(d) None of these.

2+ x dx = 2− x

2

0

∫

(d) None of these.

dx =

(a) π + 2 26.

1+ e  1 log   + −1  2e  e

e x ( log sin x + cot x ) dx =

(a) eπ / 4 log 2 1/ 2

(c)

sin x + cos x dx = 9 + 16 sin 2 x

π /4

(a)

1+ e  1 (b) log   − +1  2e  e

(b) π +

3 2

(c) π + 1

(d) None of these.

(c) 2

(d)

dx = 1 + sin x

(a) 0

(b)

1 2

3 . 2

27.

∫

2π

1 + sin

0

x dx = 2

(a) 0 28.

1

∫ cos 0

−1

∫

π /2

0

(a) 30.

∫

π 1 + log 2 4 2

(d) None of these.

(b)

π + log 2 4

(c)

π 1 − log 2 4 2

(d)

π − log 2 . 4

2

1 (π + 16 ) 36

π /2

0

∫

0

∫

0

∫

0

(c)

1 (π 2 − 16 ) 36

(d)

1 (π 2 + 16 ) . 36

(b)

π 4

(c)

π 6

(d)

π . 8

(b)

2 7

(c) 1

(d) None of these.

(b)

1 6

(c) 2

(d)

(c) log (8 × 9 )

(d) None of these.

sin x dx = cos 3 x

2 3

π /2

1 (π − 16 ) 36

tan 6 x sec2 x dx =

1 7

π /6

(b)

sin x cos x dx = 1 + sin 4 x

π 2

π /4

(a) 34.

(c) 2

0

(a) 33.

(b) 1

cos x dx = 1 + cos x + sin x

π /6

(a) 32.

(d) 4.

∫ (2 + 3x ) cos 3xdx = (a)

31.

(c) 8

x dx =

(a) 0 29.

(b) 2

1 . 3

sin x cos x dx dx = cos 2 x + 3cos x + 2

(a) log   9 8

  (b) log   8 9

35. The value of the definite integral (a) sin α

∫

1

0

dx for 0 < α < π is equal to x + 2 x cos α + 1 2

(b) tan −1 (sin α )

(c) α sin α

(d)

α −1 (sin α ) 2

36. The integral

2  −1 x −1 x + 1  tan tan +   dx = ∫−1  x2 + 1 x  3

(a) π

(b) 2π

37. If I1 = ∫

(a) I1 = I 2

∫

π /2

−π / 4

(b) I1 > I 2

∫ (2 + cos x)

∫

π

0

(a) 41.

2

0

(a) 40.

(b) −

1+ 2cos x

π /2

π 2

2 −π / 4 e 2

(c) − 2 (e −π / 4 + e −π / 4 )

(d) 0.

(b) π

(c)

1 2

(d) None of these.

(c)

π 1− a2

(d) None of these.

(c)

11 10

(d) 2.

(c)

π 2

(d)

(c)

1 log 2 3

(d) None of these.

(c)

π 4

(d) tan −1 e +

dx = 1 − 2a cos x + a 2

π 2 (1 − a 2 )

∫ (1 − x ) 1

0

∫

(d) None of these.

dx =

9

(b) π (1 − a 2 )

dx =

(a) 1 42.

(c) I1 < I 2

e− x sin xdx =

1 (a) − e−π / 2 2

39.

(d) None of these.

x 2 e dx and I 2 = ∫ dx, then 1 x log x

e2

e

38.

(c) 3π

π /3

0

(b)

1 10

cos 3xdx =

(a) π 43. The value of

(b) 0 π /4

∫

0

1 + tan x dx is 1 − tan x

1 (a) − log 2 2

44. The value of

(b)

∫

1

0

 1− e  (a) tan −1   1+ e 

π . 4

1 log 2 4

dx is e + e− x x

 e −1  (b) tan −1    e +1

π . 4

45.

∫

1 + log x dx = x

e

1

3 2

(a)

∫

46. If

(b)

∫

0

∫

3 3 (b) a = , b = − 4 4

2 2 3

π /4

0

(b)

49.

4 2 3

∫ ( x + 1)

3

0

(a)

e 4

(b)

1 32 log 4 17

1/ 2

1/ 4

dx x − x2

8 2 3

(d) None of these.

(c) π

52. The value of

(

(d) None of these.

e −1 4

(c)

e +1 4

(d) None of these.

∫ φ ( x ) dx = 2

1

(b)

1 32 log 2 17

(c)

1 16 log 4 17

(d) None of these.

(b)

π 2

(c)

π 3

(d)

=

(a) π

∫

2

3

0

π . 6

x

x

dx is

)

2 (a) 3 2 −1 log 3 53.

(d) a = b .

dx =

4 50. If x ( x + 1)φ ( x ) = 1, then

∫

3 3 (c) a = , b = 4 2

(c)

(b) π / 2

e x ( x − 1)

(a)

51.

(d) None of these.

4 sin 2θ dθ = sin 4 θ + cos 4 θ

(a) π / 4 1

1 e

dx = 1+ x − x

1

0

(a) 48.

(c)

 x 2 x log 1 +  dx = a + b log   , then  2 3

1

3 3 (a) a = , b = 2 2

47.

1 2

3

(b) 0

2 2 (c) lo g 3

(d)

(b) 2

(c) -2

(d) 1.

2π

∫ (sin x + cos x ) dx = 0

(a) 0

2

2

.

54.

∫

π /4

0

sec x is equal to 1 + 2sin 2 x

)

π   2 2

)

π  2 2 

(a) 3 log ( 2 + 1) + 2 2  

(b) 3 log 

1

(

2 +1 −

π   2 2

(d) 3 log 

(

2 +1 +

π 

1

(c) 3 log ( 2 + 1) −  

55. The value of

∫

π /2

0

sin x dx is 1 + cos 2 x

(a) π / 2

(b) π / 4

56. The value of

∫

2

1

57. The value of

(b) log 4

∫

3

(a)

∫

sin 2 x

0

π 2

(c) log 4 / e

(d) log 2 .

x2 dx is x2 − 4

 15  (a) 2 − log e   7

58. The value of

(d) π / 6 .

log x dx is

(a) log 2 / e 5

(c) π / 3

 15  (b) 2 + log e   7

sin −1 tdt + ∫

cos 2 x

0

(c) 2 + 4 log e 3 − 4log e 7 + 4 log e 5

cos −1 tdt

(b) 1

(c)

π 4

(d) None of these.

1 1 59. If for non-zero x , af ( x ) + bf   = − 5, where a ≠ b, then x x

(a)

(a

2

(c)

(a

2

1 7   a log 2 − 5a + b  2  + b 2 ) 

(b)

(a

2

1 7   a log 2 − 5a − b  2  − b 2 ) 

(d)

(a

2

60. If I n = ∫

π /4

0

61.

1 4

∫

dx = 4 + 9x2

0

(a)

π 12

62. The value of (a)

∫ f ( x ) dx = 2

1

1 7   a log 2 − 5a + b  2  − b 2 )  1 7   a log 2 − 5a − b  . 2  + b2 ) 

tan n θ dθ , then I 8 + I 6 equals

(a)

2/3

(d) 2 − tan −1 

∫

1

0

1 (3π − 4 ) 6

(b)

1 5

(c)

1 6

(d)

1 . 7

(b)

π 24

(c)

π 4

(d) 0.

1 (3 − 4π ) 6

(c)

1 (3π + 4 ) 6

(d)

x4 + 1 dx is x2 + 1 (b)

1 (3 + 4π ) . 6

15  . 7

63.

∫

a

x 2 sin x3dx equals

0

(a) (1 − cos a 3 ) 64.

65.

∫

(

π /4

0

∫

1 − x equals dx 1+ x π 

(b)

∫

1

∫

1

2

∫

π /2



π 2

π 2

(c)

(b) 0

(c) 1

(d) 2π .

(d) (π + 1) .

(d) log (1 + e ) .

1 2 (log x ) 2

(c)

log x 2 2

(d) None of these.

(c)

π ab

(d)

dx = a cos x + b 2 sin 2 x 2

0

∫

(c)

(b)  2 + 1  

(b)

2

(a) π ab 69.

1 (1 − cos a3 ) 3

log x 2 dx = x

x

(a) ( log x ) 68.

(d)

1 dx is equal to x

e

(a) ∞ 67.

π 2

π

 

(a)  2 − 1 66.

1 1 − cos a 3 ) ( 3

)

2π

0

(c) −

tan x + cot x dx equals

(a) 1

(b) 3 (1 − cos a 3 )

π /4

(b) π 2 ab 5π / 4

π . 2ab

π /4

(cos x − sin x ) dx + ∫π / 4 (sin x − cos x ) dx + ∫2π (cos x − sin x ) dx is equal to

0

2 −2

(a)

(b) 2 2 − 2

(c) 3 2 − 2

(d) 4 2 − 2 .

(b) −2 ≤ a ≤ 4

(c) −2 ≤ a ≤ 0

(d) a ≤ −2 or a ≥ 4 .

(b) π / 4

(c) π / 2

(d) −π / 4 .

(c) π / 4

(d) π / 3 .

(∫ x dx ) ≤ (a + 4), then a

70.

0

(a) 0 ≤ a ≤ 4 71.

∫

dx = x + 2x + 2

0

2

−1

(a) 0 72.

∫

1

3

1 dx is equal to 1 + x2

(a) π /12

(b) π / 6

73.

∫

3

( x − 1)( x − 2)( x − 3)dx =

1

(a) 3 74.

∫

dx

∫ ( x − 3) 15

8

1

0

∫

π

∫

3

0

(d) log (8 / 3) .

(c)

(b) 3 / 8

1 3 log 2 5

(d)

1 3 log . 5 5

(c) 4 / 3

(d) π .

(c) π / 2

(d) π .

 1+ x  sin  2 tan −1  dx =  1 − x  

(b) π / 4

3x + 1 dx = x2 + 9

(

)

(a) log 2 2 +

(

π 12 dx

2

∫ x (1 + x )

1 17 log 4 32

(b)

80. The value of

4

1

∫

3

2

(a) 2 log 2 −

1 6

81. The value of

∫

1

(

)

(c) log 2 2 +

π 6

(

1 17 log 4 2

(c) log

17 2

(d)

1 32 log . 4 17

(c) log

4 1 − 3 6

(d) log

x +1 dx is x ( x − 1) 16 1 − 9 6

16 1 + . 9 6

log x dx is (b) 1

82. The value of I = ∫

π /2

0

(sin x + cos x )

(c) e − 1

(d) e + 1 .

(c) 2

(d) 0.

2

1 + sin 2 x (b) 1

)

(d) log 2 2 +

2

(a) 0

(a) 3

π 2

is

(b) log e

)

(b) log 2 2 +

79. The value of (a)

1 5 log 3 3

sin 3 θ dθ is

0

(a) π / 6 78.

(c) log ( 4 / 3)

(b)

(a) 0

∫

(b) log (1/ 4 ) =

x +1

1 5 log 2 3

76. The value of

77.

(d) 0.

2

2

(a)

(c) 1

dx = x −x

3

(a) log ( 2 / 3) 75.

(b) 2

dx is

π 3

83.

∫

π /8

2 3

(a) 84.

cos 3 4θ d θ =

0

(b)

8

2 − 3x

3

x (1 + x )

∫

(c)

∫

1

(b) log (3 / e 3 )

(b) e + 2 dx

2

1 + x2

1

and I 2 = ∫

2

1

(a) I1 > I 2

∫

(c) 4 log (3 / e3 )

(d) None of these

(c) e 2 − 2

(d) e 2 .

(c) I1 = I 2

(d) I1 > 212 .

(c) 0

(d) None of these.

dx then x

(b) I 2 > I1

87. The value of

1 (d) . 6

x 2 e x dx is equal to

0

(a) e − 2 86. Let I1 = ∫

1 3

dx is equal to

(a) 2 log (3 / 2e3 ) 85. The value of

1 4

tan x

1/ e

cot x t dt dt +∫ = 2 I /e 1+ t t (1 + t 2 )

(a) -1

(b) 1

3π / 4

88.

dx x is equal to π / 4 1 + cos

∫

(a) 2

(b) -2

89. The value of

∫

e2

1

dx x (1 + ln x )

2

(a) 2 / 3 90.

∫

π /2

π /4

∫

du

x

log 2

(e

u

− 1)

1/ 2

=

1 (d) − . 2

is

(b) 1/ 3

(c) 3/ 2

(d) ln 2.

(b) 1

(c) 0

(d)

(c) 4

(d) -1.

1 . 2

π , then e x = 6

(a) 1 92. If g (1) = g ( 2 ) , then (a) 1

1 2

cos ec 2 xdx =

(a) -1 91. If

(c)

(b) 2

∫ ( fg ( x )) 2

1

(b) 2

−1

f ' ( g ( x ) ) g ' ( x ) dx is equal to (c) 0

(d) None of these.

L E V E L - 2 ( Properties of Definite Integration )

∫

1.

π

0

xf (sin x ) dx = π

(a) π ∫ f (sin x ) dx 0

∫

2.

π /2

(a) π

∫

π /2

0

x

a

(a) e x ( x 3 + 3 x 2 ) 1

−1

∫

π /2

0

(a) 7.

∫

π /2

0

(d) None of these.

π 2

(c)

π 4

(d)

π . 3

(b)

π 2

(c)

π 3

(d)

π . 4

d f (x) = dx

3 x (b) x e

3 a (c) a e

(d) None of these.

(b) 0

(c) 2

(d) -2.

(c) π log e 2

(d) 0.

x x dx =

(a) 1 6.

π π /2 f (sin x ) dx 2 ∫0

(b)

4. If f ( x ) = ∫ t 3et dt , then

∫

(c)

dθ = 1 + tan θ

(a) π

5.

π π f (sin x ) dx 2 ∫0

cot x dx = cot x + tan x

0

3.

(b)

log tan x dx =

π log e 2 2

(b) −

π log e 2 2

log sin x dx =

1 π  (a) −   log 2 (b) π log 2 2 π / 2 cos x − sin x dx = 8. ∫0 1 + sin x cos x

(a) 2

(b) -2

(c) − π log

(c) 0

1 2

(d)

π log2 . 2

(d) None of these.

9.

∫

1

−1

 2−x  log   dx =  2+ x 

(a) -2 10.

∫

1

−1

∫

0

∫

π /4

0

(d) 2.

(c) π / 2

(d) π / 4 .

log (1 + tan θ ) dθ = (b)

0

(a) 1 14.

(c) 0

π 1 log 4 2

(c)

π log 2 8

(d)

π 1 log . 8 2

(c)

π 4

(d) 0.

sin 2θ dθ = a − b cos θ

2π

∫

(b)-1

(b) π

π log 2 4

(a) 13.

(d) 0.

sin 3/ 2 xdx = cos3/ 2 x + sin 3/ 2 x

π /2

(a) 0 12.

(c) -1

x17 cos 4 xdx =

(a) -2 11.

(b)1

(b) 2

∫ f (1 − x )dx has the same value as the integral 1

0

∫

(a)

1

0

f ( x )dx

(b)

∫

1

0

f ( − x )dx

(c)

∫

1

0

f ( x − 1)dx

(d)

∫ f ( x )dx . 1

−1

  1 − x  ∫ (cos x ) log  1 + x  dx = 1/ 2

15.

−1/ 2

(a) 0 16. The value of (a) 17. If

(b) 1 1

dx

0

x + 1− x2

∫

π 3

∫

1

−1

(d) 2e1/ 2 .

is

π 2

(c)

1 2

(d)

π . 4

∫ f ( x ) = 0, then 1

−1

(a) f ( x ) = f ( − x ) 18.

(b)

(c) e1/ 2

(b) f ( − x ) = − f ( x )

(c) f ( x ) = 2 f ( x )

(d) None of these.

(b) 0

(c) 2

(d) 4.

1 − x dx =

(a) -2

19.

20.

∫

π

0

x sin 3 x dx =

(a)

4π 3

∫

1 − x 2 dx =

2

−2

(a) 2 21.

∫

(b) 4

π /2

∫

π /2

0

(b)

(b)

(a) 2 +

(c)

π 4

(d) None of these.

π 8

(c)

π2 8

(d)

π /2

0

π2 . 16

π  sin  x −  dx is 4 

(c) −2 + 2

(b) 2 − 2

2

(d) 0.

a

0

∫ f ( a + x ) dx a

∫

π /2

0

∫

π

0

(

(c)

0

(b) 2

∫ f ( x − a ) dx a

0

(d)

∫ f ( a − x ) dx . a

0

)

2 −1

2 −1

(d) 2

(

)

2 +1 .

cos x dx = (b) 0

27. The value of the integral (a) 3/2 1.5

0

(c)

sin x − cos x dx =

(a) π

∫

∫ f (2a + x ) dx a

(b)

0

(a) 0

28.

(d) 8.

∫ f ( x ) dx = (a)

26.

(c) 6

π 2

∫

23. The correct evaluation of

25.

(d) None of these.

x sin x cos x dx = cos 4 x + sin 4 x

(a) 0

24.

(c) 0

cos x dx = sin x + cos x

0

(a) 2 22.

2π 3

(b)

∫

π /4

−π / 4

(c) 2

(d) 1.

(c) 3/8

(d) 8/3.

sin −4 x dx =

(b) -8/3

 x 2  dx, where [ . ] denotes the greatest integer function, equals

(a) 2 + 2

(b) 2 − 2

(c) −2 + 2

(d) −2 − 2 .

29.

∫

π

0

x tan x dx = sec x + tan x

π −1 2

(a) 30.

∫

π

0

∫

1

−1

(c)

π2 (b) 2

3π 2 (c) 2

(b) 1

(c)

∫

32. For any integer n , the integral (a) -1

∫

e

1/ e

π2 (d) . 3

π

0

1 2

(d) 2.

esin x cos3 ( 2n + 1) x dx = 2

(b) 0

(c) 1

(d) π .

 1 (b) 2 1 −   e

(c) e −1 − 1

(d) None of these.

log x dx =

(a) 1 − 34.

π  (d) π  − 1 . 2 

sin 3 x cos 2 xdx =

(a) 0

33.

π +1 2

x tan x dx = sec x + cos x

π2 (a) 4 31.

π  (b) π  + 1 2 

1 e

∫ ( x − [sin x ]) dx is equal to (where [.] represents greatest integer function) π /2

0

π2 (a) 8

π2 −1 (b) 8

π2 −2 (c) 8

(d) None of these.

n

35. The value of the integral I = ∫ x (1 − x ) dx is 1

0

(a)

1 n +1

(b)

36. The value of

(c)

(c) −

∫

π /2

0

∫

3π / 4

π /4

5π 3

(d)

5π . 3

(d)

π . 4

dx is 1 + tan 3 x

(b) 1

π 8

1 1 + . n +1 n + 2

π

(a) 0 38. The value of

(d)

∫ [2sin x ]dx , where [ . ] represents the greatest integer function, is (b) −2π

37. The value of

1 1 − n +1 n + 2

2π

(a) −π

(a) π tan

1 n+2

(c)

π 2

φ dφ , is 1 + sin φ

(b) log tan

π 8

(c) tan

π 8

(d) None of these.

39. If f ( a + b − x ) = f ( x), then ∫ x f ( x ) dx = b

a

(a) 40.

∫

π

0

a+b b f (b − x ) dx 2 ∫a

∫

0

π /4

(b) f ( 2a − x ) = f ( x )

(c) f ( a − x ) = − f ( x )

sin 2 x dx and J = ∫

(b)

∫

5 6

2

(b) 0

∫

π

0

(b) -1

(c) 0

(d) None of these.

1 . 2

1 dx = 1 + tan x

π 4

(c)

π 6

(d) 1.

1

(a) 0

(a)

(d)

2

sin x − x 2 ∫−1 3 − x dx is

−1

(c) -1

ecos x cos5 3x dx is

47. The value of

∫

(d) 12.

x dx is 5− x + x

3

(b)

1

(c) 21

1

π 2

48.

(d)

∫ ( x − 3 + 1 − x ) dx is

(a) 1

(a)

J . 2

(c) J

5

(a) 1 45. The value of

(d) f ( a − x ) = f ( x ) .

cos 2 x dx, then I =

(b) 2J

(a) 10 44. The value of

π /4

0

π −J 4

π /2

(d) π 2 .

0

43. The value of

0

(d) None of these.

a

0

∫

b−a b f ( x ) dx 2 ∫a

f ( x ) dx = 2 ∫ f ( x ) dx , then

2a

42. If I = ∫

46.

(c)

(c) 1

(b) 0

(a) f ( 2a − x ) = − f ( x )

(a)

a+b b f ( x ) dx 2 ∫a

x sin x dx =

(a) π 41. If

(b)

sin x dx 0 3− x

(b) 2∫

1

2 1 (c) 2 ∫ − x dx 0 3− x

sin x − x 2 dx . 0 3− x

(d) 2 ∫

1

sin11 xdx is equal to 10 8 6 4 2 . . . . 11 9 7 5 3

(b)

10 8 6 4 2 π . . . . . 11 9 7 5 3 2

(c) 1

(d) 0.

49. To find the numerical value of (a) p 50.

∫

1

−1

∫

−2

2

+ qx + s ) dx, it is necessary to know the values of constants

(b) q

(c) s

(d) p and s.

(b) 1

(c) 0

(d) π .

(b) 0

(c) -1

(d) None of these.

 1+ x  log   dx =  1− x 

(a) 2 51.

∫ ( px 2

π /2

−π / 2

cos x dx = 1 + ex

(a) 1

52. If [ x ] denotes the greatest integer less than or equal to x , then the value of the integral (a) 5/3 53.

∫

π

0

∫

π

0

(c) 8/3

(d) 4/3.

(b) 0

(c) 1

(d) π .

(b) π log e 2

(c)

55. If f ( x ) is an odd function of x , then (a) 0

∫

π

0

(b)

58.

∫

π /2

0

(b)

π 2

∫

x tan −1 x dx equals

−1

π 2 0

∫

π 2 π − 2

∫

f (cos x ) dx

π 1 log e   2 2

(d) None of these.

f (cos x ) dx is equal to π

(c) 2 ∫ 2 f (sin x ) dx

(d)

0

π 2

(c) 0

π

∫ f (cos x ) dx . 0

(d) None of these.

sin x dx equals sin x + cos x

(a) 1

x 2 [ x ] dx equals

sin 2 x dx is equal to

(a) π 57.

0

log sin 2 x dx =

1 (a) 2π log e   2

56.

2

cos3 x dx =

(a) -1 54.

(b) 7/3

∫

π  (a)  − 1  2 

(b)

π 3

π  (b)  + 1 2 

(c)

π 4

(c) (π − 1)

(d)

π . 6

(d) 0.

59.

∫

a

−a

sin x f (cos x ) dx =

(a) 2∫0 sin xf ( cos x ) dx a

60. The value of

∫

2π

0

∫

2

∫

3

0

(b) 3 / 8

(c) 8 / 3

(d) π .

(b) 1 / 2

(c) 3 / 2

(d) 7 / 2.

(b) 5 / 2

(c) 3 / 2

(d) -3 / 2.

(c) π

(d) 2π .

(c) 3 / 7

(d) 5 / 6.

(c) 0

(d) None of these.

2 − x dx equals

(a) 2 / 7 63. The value of (a)

∫

π /2

0

2sin x dx is 2sin x + 2cos x

π 4

(b)

64. The value of

∫

1

0

(b) 4 / 3 3

π /2

2 sin x e − cos x dx is equal to 2 −π / 2 1 + cos x

∫

(a) 2e −1

(b) 1

66. f ( x ) = f ( 2 − x ) , then (a) 67.

π 2

3x 2 − 1 dx is

(a) 0 65.

(d) None of these.

x dx

−1

(a) 5 / 2 62.

(c) 1

sin 3 θ dθ is

(a) 0 61.

(b) 0

∫

∫ f ( x ) dx 1

(b)

0

ex

π /2 0

e

x2

(a) π / 4

+e

∫

1.5

0.5

∫

xf ( x ) dx equals

1.5

0.5

f ( x ) dx

(c) 2∫

1.5

0.5

f ( x ) dx

(d) 0.

2

π   2 −x  

2

d x is

(b) π / 2

(c) eπ /16 2

(d) eπ / 4 . 2

68. If [ x ] denotes the greatest integer less than or equal to x , then the value of (a) 1

(b) 2

(c) 4

∫

(d) 8.

5

1

 x − 3  dx is

69.

∫

2

−2

| x | dx =

(a) 0

(b) 1

(c) 2

(d) 4.

70. Suppose f is such that f ( − x ) = − f ( x ) for every real x and (a) 10

(b) 5

71. Let I1 = ∫

π −a

a

72.

∫

1+ x  cos x.ln   dx is equal to 1− x 

1/ 2

−1/ 2

(b) 1

73. The value of

∫

e2

e

−1

3 2

0

0

−1

(d) -5.

f (sin x ) dx, then I 2 is equal to

(b) π I1

(a) 0

(a)

π −a

a

π I1 2

1

(c) 0

xf (sin x ) dx, I 2 = ∫

(a)

∫ f ( x ) dx = 5, then ∫ f (t ) dt =

log e x dx is x 5 (b) 2

(c)

2 I1 π

(d) 2I1 .

(c) 2

(d) ln 3.

(c) 3

(d) 5.

3 ecos x sin x, x ≤ 2 , then ∫ f ( x ) dx is equal to −2 2, otherwise 

74. If f ( x ) =  (a) 0

(b) 1

(c) 2

(d) 3.

75. If f : R → R and g : R → R are one to one, real valued functions, then the value of the integral

∫ ( f ( x ) + f ( − x )) ( g ( x ) − g ( − x )) dx is π

−π

(a) 0 76.

π /3

dx

π /6

1 + cot x

∫

∫

π /2

0

∫

1

−1

(

(d) None of these.

(b) π / 6

(c) π /12

(d) π / 2 .

(c) 3π / 4

(d) π .

sin 2 / 3 x dx is sin 2 / 3 x + cos 2 / 3 x

(a) π / 4 78.

(c) 1

is

(a) π / 3 77. The value of

(b) π

(b) π / 2

)

log x + x 2 + 1 dx =

(a) 0

(b) log 2

(c) log

1 2

(d) None of these.

79. The value of the integral (a) −π 80.

∫

0

2a

0

(b) 2

(b) 0

0

π

0

∫

0

(a) 2 − 2 1000

0

a

0

0

(d)

∫ f ( x ) dx + ∫ a

2a

0

0

e

x −[ x ]

∫

9

0

2

 x + 2  dx, where [ . ] is the greatest integer function  

π /2

(c) 23

(d) None of these.

 x 2  dx, where [ . ] is the greatest integer function

2 −1

(d)

(c) 1000 (e − 1)

(d)

(c)

2 −2.

dx is (b)

86. The value of the ingral (a)

(d) π .

(c) 1

(b) 2 + 2

(a) e1000 − 1

0

∫ f ( x ) dx + ∫ f (2a − x ) dx

(b) 22

84. The value of

∫

(c)

(b) 0

(a) 31

87.

(d) -1.

2

83. Find the value of

∫

(c) 1

esin x cos3 x dx is equal to

(a) -1

85.

(d) 2 π .

a

a

∫

(c) π

f ( x ) dx =

(a) 2∫ f ( x ) dx 82.

dx, ( a and b are int eger ) is

1 + cos 2 x dx is equal to 2

π

∫

2

−π

(b) 0

(a) 0 81.

π

∫ (cos ax − sin bx )

a 2

e1000 − 1 e −1

an −1 n 1 n

∫

(b)

e −1 . 1000

x dx is a−x + x

na + 2 2n

(c)

na − 2 2n

(d) None of these.

sin 2 x log tan x dx is equal to

(a) π

(b) π / 2

88. The integral (a) −

1 2

(c) 0

(d) 2π .

  1+ x   x ] + log  [   dx equal ( where [.] is the greatest integer function )  −1/ 2  1− x   

∫

1/ 2

(b) 0

(c) 1

1 (d) 2 log . 2

f ( 2a − x ) dx .

∫ (sin x + sin x ) dx = 2π

89.

0

(a) 0

(b) 4

(c) 8

(d) 1.

(c) 0

(d)

(c) 1/8

(d) None of these.

(b) 12

(c) 9

(d) 18.

(b) 2

(c) 1/2

(d) 1.

∫ (3sin x + sin x ) dx is π /2

90. The value of

3

−π / 2

(a) 3

(b) 2

91. The value of I = ∫0 x x − 1

(a) 1/3

∫

8

0

x − 5 dx

(a) 17 93.

∫

2

0

x − 1 dx =

(a) 0 94.

∫ [ x] dx = 2

−2

(where [.] denotes greatest integer function)

(a) 1 1

95.

∫ tan 0

1 dx is 2

(b) 1/4

92. The value of

10 . 3

−1

(b) 2

(c) 3

(b) − ln 2

(c)

(d) 4.

 1   x2 x 1 dx  − + 

(a) ln 2 96. The value of

∫

b

a

π + ln 2 2

(d)

π − ln 2 . 2

x dx, a < b < 0 is x

(a) − ( a + b )

(b) b − a

(c) a − b

(d) a + b .

−2    1+ x   1− x  + qln  + r  dx depends on 97. The value of ∫−2  pln      1− x   1+ x    2

(a) The value of p 98.

∫

π

0

(b) The value of q

(c) The value of r

(d) The value of p and q.

(c) π

(d) None of these.

xdx 1 + sin x is equal to

(a) −π

(b)

π 2

99. The value of (a)

∫

3

−2

1 − x 2 dx is

1 3

14 3

(b)

(c)

7 3

(d)

28 . 3

100. If f ( x ) = x − 1 , then ∫ f ( x ) dx is 2

0

(a) 1

(b) 0

∫

101. If

π

0

xf (sin x ) dx = A∫

π /2

0

(d) -2.

f (sin x ) dx, then A is

(b) π

(a) 2π 102.

(c) 2

(c)

π 4

(d) 0.

π /2

∫ (sin x − cos x ) log (sin x + cos x ) dx = 0

(a) -1

(b) 1

103. The function L ( x ) = ∫1

x

(c) 0

(d) None of these.

dt satisfies the equation t

x (a) L ( x + y ) = L ( x ) + L ( y ) (b) L   = L ( x ) + L ( y ) (c) L ( xy ) = L ( x ) + L ( y ) (d) None of these.  y

∫e 1

104. The value of integral (a) ( 0,1 ) 105. If P = ∫

x2

0

dx lies in the interval

(b) ( -1,0 ) 3π

0

(c) ( 1, e )

(d) None of these.

f (cos2 x ) dx and Q = ∫ f (cos2 x ) dx, then π

0

(a) P - Q = 0

(b) P - 2Q = 0

(c) P - 3Q = 0

(d) P - 5Q = 0.

106. Let a, b, c be non - zero real numbers such that

∫ (3ax 3

0

2

+ 2bx + c ) dx = ∫ (3ax 2 + 2bx + c ) dx, then 3

1

(a) a + b + c = 3 107.

π

(b) a + b + c = 1

∫ (cos px − sin qx )

2

−π

(a) −π

(c) a + b + c = 0

(d) a + b + c = 2.

dx is equal to ( where p and q are integers )

(b) 0

(c) π

(d) 2π .

(c) g ( x ) g (π )

(d) g ( x ) / g (π ) .

108. If g ( x ) = ∫ cos4 t dt , then g ( x + π ) equals x

0

(a) g ( x ) + g (π )

(b) g ( x ) − g (π )

∫ (1 + e 1

109. The value of

0

(a) -1 110.

∫

(b) 2

(a) a

(b)

111. The value of

∫

nπ +υ

0

112. If un = ∫

π /4

0

(c) 2a

(d) 0.

(c) 2n + 1

(d) 2n + cos υ .

1 n +1

(c)

1 2n − 1

(d)

(b) log 2

(c)

π log 2 2

(d) −

(b) π log 2

(c) − π log 2 2

(d) − π log 2 .

(c) π log 2

(d) − π log 2 .

(b)

1 . 2n + 1

π 

∫ log sin  2 x  dx = 0

1

log x

0

1 − x2

∫

(a)

∫

π /2

0

x cot x dx equals

π log 2 2

(b)

∫ (x 0

116. The integral value (a) 2 117. If

∫

∫

2 nπ

0

(a) n

π log 2 2 3

−2

+ 3 x 2 + 3 x + 3 + ( x + 1) cos ( x + 1)) dx is

(b) 4 1

sin x

(a) 3

π log 2 . 2

dx =

π log 2 2

(a) −

118.

(d) None of these.

tan n x dx, then un + un − 2 =

(a) − log 2

115.

a 2

(b) 2n + 1 − cosυ

1 n −1

1

114.

(c) 1 + e−1

sin x dx is

(a) 2n + 1 + cos υ

113.

) dx =

f (x) dx = f ( x ) + f ( 2a − x )

2a

0

(a)

− x2

(c) 0

(d) 8 .

 1   π t 2 f (t ) dt = 1 − sin x, x ∈  0,  then f   equal to  2  3

(b)

1 3

(c)

1 3

(d)

3

1    sin x − 2 sin x  dx equals  

(b) 2n

(c) - 2n

(d) None of these

119. The value of

1 dx is − a x + x3

∫

a

(a) 0

120.

(b)

π /3

dx

π /6

1 + tan x

∫

∫

a

0

1 dx 1 + x6

(c) 2 ∫0

a

1 dx 1 + x3

(d)

∫

1

a

0

1 + (a − x )

3

=

(a) π /12

(b) π / 2

(c) π / 6

(d) π / 4

(c) 3π / 2

(d) π

sin 4 x dx = 121. ∫−π sin 4 x + cos 4 x π

(a) π / 4

(b) π / 2

122. If f is continuous function, then f ( x ) dx = ∫  f ( x ) − f ( − x ) dx 0

(a)

∫

(c)

∫ f ( x ) dx = ∫ f ( x − 1) dx

2

2

−2 5

4

−3

−4

5

2 f ( x ) dx = ∫ f ( x − 1) dx 10

(b)

∫

(d)

∫ f ( x ) dx = ∫ f ( x − 1) dx

−3

−6

5

6

−3

−2

n n 1   n + + + ..... +  is equal to 123. The value of lim  2 2 n →∞ 1 + n 2 4+n 9+ n 2n  

(a)

π 2

124. lim n →∞ (a)

(b)

π 4

(c) 1

(d) None of these.

1 4 1 + 3 + ..... + is equal to 3 3 1 +n 2 +n 2n 3

1 log e 3 3

(b)

1 log e 2 3

(c)

1 1 log e 3 3

(d) None of these.

(c)

1 99

(d)

199 + 299 + 399 + ....n 99 = 125. lim n →∞ n100 (a)

9 100

(b)

1 100

1 . 101

1/ n

 n!  126. lim n →∞  n n   

(a) e 127. lim n →∞

equals (b) 1/ e

(c) π / 4

(d) 4 / π .

(c) −1 + 2

(d) 1 + 2 .

r 1 2n equals ∑ 2 n r =1 n + r 2

(a) 1 + 5

(b) −1 + 5

dx .

  + + + .....  = 128. lim n →∞  n 1 2 2 n n n + +   1

1

1

(a) 0

1

(b) log e 4

(c) log e 3

(d) log e 2 .

(c) π / 4

(d) π / 2 .

n

k 2 is equal to k =1 n + k

129. lim ∑ n →∞ (a)

2

1 log 2 2

(b) log 2

1

130. lim  + n →∞

n 

1 n2 + n

+

(a) 2 + 2 2

1

+ ..... +

n 2 + 2n

  is equal to n 2 + ( n − 1) n  1

(b) 2 2 − 2

(c) 2 2

(d) 2.

1p + 2 p + 3 p + ...... + n p = 131. lim n →∞ n p +1 (a)

1 p +1

(b)

1 1− p

(c)

1 1 − p p −1

1 . p+2

(d)

r

n 1 132. lim ∑ e n is equal to n →∞ r =1 n

(a) e + 1

(b) e − 1

133. The correct evaluation of (a)

8π 3

(b)

∫

π

0

(c) 1- e

(d) e.

sin 4 x dx is

2π 3

(c)

4π 3

(d)

3π . 8

134. The points of intersection of F1 ( x ) = ∫ ( 2t − 5 ) dt and F2 ( x ) = ∫ 2tdt , are  6 36  (a)  ,   5 25 

135.

∫

b −c

0

x

x

2

0

2 4 (b)  ,  3 9

1 1 (c)  ,  3 9

1 1  (d)  ,  .  5 25 

f " ( x + a ) dx =

(a) f ' ( a ) − f ' (b )

(b) f ' (b − c + a ) − f ' ( a )

(c) f ' (b + c − a ) + f ' ( a )

(d) None of these.

  136. The greatest value of the function F ( x ) = ∫ t dt = on the interval  − ,  is given by 1  2 2 1 1

x

(a) 137.

∫

3 8

π /2

−π / 2

(a)

(b) −

1 2

3 8

(d)

2 . 5

6 15

(d)

8 . 15

(c) −

sin 2 x cos2 x (sin x + cos x ) dx =

2 15

(b)

4 15

(c)

∞

∫

138.

(x +

0

dx x +1 2

)

3

=

3 8

(a)

1 8

(b)

(c) −

3 8

(d) None of these.

139. If f ( x ) = 2 e − t dt , then f ( x ) increases in ∫ x 2 +1

2

x

(a) ( 2, 2 )

(b) No value of x

∫ f ( x ) dx = xe

40. If

− log x

(a) 1

∫ ( π /2

0

(a) 2/9 142.

∞

∫

0

∫

∞

0

(c) ce x

(d) log x .

(c) 8/45

(d) 5/2.

(c) (π / 2 ) log 2

(d) − (π / 2 ) log 2 .

(c) ∞

(d) None of these.

(c) -1

(d) 1.

)

3

sin θ cos θ dθ is

(b) 2/15

1  dx  log  x +  is equal to x  1 + x2 

(a) π log 2 143.

(d) ( −∞, 0 )

+ f ( x ) , then f ( x ) is

(b) 0

141. The value of

(c) (0, ∞ )

(b) −π log 2

x ln x dx

is equal to

(1 + x )

2 2

(a) 0

(b) 1

144. If f (t ) = ∫−t t

dx , then f ' (1) is 1+ x2

(a) Zero

(b) 2 / 3

145. If F ( x ) = ∫ 2 log t dt , ( x > 0 ) , then F ′ ( x) = x3

x

(a) (9 x 2 − 4 x ) log x 146.

∫

1

0

(b) ( 4 x − 9 x 2 ) log x

(c) (9 x 2 + 4 x ) log x

(d) None of these.

d  −1  2 x   sin  dx is equal to 2  dx   1 + x 

(b) π

(a) 0 147. Let f ( x ) = ∫

x

1

(a) ±1

(c) π / 2

(d) π / 4 .

2 − t 2 dt. Then real roots of the equation x 2 − f ' ( x ) = 0 are (b) ±

1 2

(c) ±

1 2

(d) 0 and 1.

∞

xdx

148. ∫ (1 + x ) (1 + x ) = 2

0

(b) π / 2

(a) 0

 esin x  d F (x) =   ; x > 0. If dx  x 

149. Let

(a) 15

(c) π / 4

∫

4

1

(d) 1.

3 sin x3 e dx = F ( k ) − F (1) , then one of the possible value of ‘k’, is x

(b) 16

(c) 63

(d) 64.

(c) x cos x

(d) None of these.

150. If f ( x ) = ∫ t sin t dt , then f ' ( x ) = x

0

(a) cos x + x sin x

(b) x sin x

1 2 4 1 1  sec 2 2 + 2 sec 2 2 + .... + sec 2 1 equals 151. lim n →∞  n 2 n n n n  

(a) tan1

(b)

1 tan1 2

(c)

1 sec1 2

(d)

1 cos ec1 . 2

152. Area bounded by the curve y = log x, x − axis and the ordinates x = 1, x = 2 is (a) log 4 sq. unit

(b) ( log 4 + 1) sq. unit

(c) ( log 4 − 1) sq. unit

(d) None of these.

153. Area bounded by the parabola y = 4 x 2 , y − axis and the lines y = 1, y = 4 is (a) 3 sq. unit

(b)

7 sq. unit 5

(c)

7 sq. unit 3

(d) None of these.

8   154. If the ordinate x = a divides the area bounded by the curve y =  1 + 2  , x − axis and the ordinates  x  x = 2, x = 4 into two equal parts, then ‘a’ =

(a) 8

(b) 2 2

(c) 2

(d)

2.

155. Area bounded by y = x s in x and x − axis between x = 0 and x = 2π , is (a) 0

(b) 2π sq.unit

(c) π sq.unit

156. Area under the curve y = s in 2 x + cos 2 x between x = 0 and x = (a) 2 sq. unit

(b) 1 sq. unit

(c) 3 sq. unit

(d) 4π sq.unit .

π , is 4 (d) 4 sq. unit .

157. Area under the curve y = 3x + 4 between x = 0 and x = 4, is (a)

56 sq. unit 9

(b)

64 sq. unit 9

(c) 8 sq. unit

(d) None of these.

158. Area bounded by parabola y 2 = x and straight line 2 y = x is (a)

4 3

(b) 1

(c)

2 3

(d)

1 . 3

159. Area bounded by lines y = 2 + x, y = 2 − x and x = 2 is (a) 3

(b) 4

(c) 8

(d)16.

160. The ratio of the areas bounded by the curves y = cos x and y = cos 2 x between x = 0, x = π / 3 and x − axis, is (a)

2 :1

(b) 1: 1

(c) 1 : 2

(d) 2 : 1.

161. The area bounded by the curve y = x 3 , x − axis and two ordinates x = 1 to x = 2 equal to (a)

15 sq. unit 2

(b)

15 sq. unit 4

(c)

17 sq. unit 2

(d)

17 sq. unit . 4

162. The area bounded by the x − axis and the curve y = sin x and x = 0 , x = π is (a) 1

(b) 2

(c) 3

(d) 4.

163. For 0 ≤ x ≤ π , the area bounded by y = x and y = x + sin x, is (a) 2

(c) 2π

(b) 4

(d) 4π .

164. The area of the region bounded by the x − axis and the curves defined by y = tan x, ( −π / 3 ≤ x ≤ / 3) is (a) log 2

(b) − log 2

(c) 2 log 2

(d) 0.

165. If the area above the x − axis , bounded by the curves y = 2kx and x = 0 and x = 2 is

3 , In 2

then the value of ‘ k ’ is 1 (a) 2

(b) 1

(c) -1

(d) 2.

166. The area bounded by the x-axis, the curve y = f ( x ) and the lines x = 1, x = b is equal to all b > 1, then f ( x ) is (a)

x −1

(b)

x +1

(c)

x2 + 1

(d)

x 1 + x2

.

b2 + 1 −

2

for

167. The area bounded by the curve y = f ( x ) , x − axis and ordinates x = 1 and x = b is (b − 1) sin (3b + 4 ) , then f ( x ) is (a) 3 ( x − 1) cos (3x + 4 ) + sin (3x + 4 )

(b) (b − 1) sin (3x + 4 ) + 3cos (3x + 4 )

(c) (b − 1) cos (3x + 4 ) + 3sin (3x + 4 )

(d) None of these.

168. The area of the region (in the square unit) bounded by the curve x 2 = 4 y, line x = 2 and x axis is (a) 1

(b)

2 3

(c)

4 3

(d)

8 . 3

169. Area under the curve y = x 2 − 4 x within the x − axis and the line x = 2 , is (a)

16 sq.unit 3

(b) −

16 sq.unit 3

(c)

4 sq.unit 7

(d) Cannot be calculated.

170. Area bounded by the curve xy − 3 x − 2 y − 10 = 0, x − axis and the lines x = 3, x = 4 is (a) 16 log 2 − 13

(b) 16 log 2 − 3

(c) 16 log 2 + 3

(d) None of these.

171. The area bounded by curve y 2 = x, line y = 4 and y − axis is (a)

16 3

(b)

64 3

(c) 7 2

(d) None of these.

172. The area bounded by the straight lines x = 0, x = 2 and the curves y = 2 x , y = 2 x − x 2 is 4 1 (a) 3 − log 2

3 4 (b) log 2 + 3

4 (c) log 2 − 1

3 4 (d) log 2 − 3 .

173. The area between the curve y = sin 2 x, x − axis and the ordinates x = 0 and x = (a)

π 2

(b)

π 4

(c)

π 8

π is 2

(d) π .

174. The area bounded by the circle x 2 + y 2 = 4, line x = 3 y and x − axis lying in the first quadrant, is (a)

π 2

(b)

π 4

(c)

π 3

(d) π .

175. The area bounded by the curve y = 4 x − x 2 and the x − axis , is (a)

30 sq.unit 7

(b)

31 sq.unit 7

(c)

32 sq.unit 3

(d)

34 sq.unit . 3

176. Area of the region bounded by the curve y = tan x, tangent drawn to the curve at x = (a)

1 4

(b) log 2 +

1 4

(c) log 2 −

1 4

π and the x − axis is 4

(d) None of these.

177. The area between the curve y = 4 + 3 x − x 2 and x − axis is (a) 125 / 6

(b) 125 / 3

(c) 125 / 2

(d) None of these.

178. Area inside the parabola y 2 = 4ax , between the lines x = a and x = 4a is equal to (a) 4a 2

(b) 8a 2

(c)

28 2 a 3

(d)

35 2 a . 3

179. The area of the region bounded by y = x − 1 and y = 1 is (a) 2

(b) 1

(c)

1 2

(d) None of these.

180. The area between the curve y 2 = 4ax, x − axis and the ordinates x = 0 and x = a is (a)

4 2 a 3

(b)

8 2 a 3

(c)

2 2 a 3

(d)

5 2 a . 3

181. The area of the curve xy 2 = a 2 ( a − x ) bounded by y − axis is (a) π a 2

(b) 2π a 2

(c) 3π a 2

(d) 4π a 2 .

182.The area enclosed by the parabolas y = x 2 − 1 and y = 1 − x 2 is (a) 1 / 3

(b) 2 / 3

(c) 4 / 3

(d) 8 / 3.

183. The area of the smaller segment cut off from the circle x 2 + y 2 = 9 by x = 1 is (a)

(

1 9sec −1 3 − 8 2

)

(c) 8 − 9sec −1 (3)

(b) 9sec −1 (3) − 8 (d) None of these.

184.The area of the region bounded by the curves y = x − 2 , x = 1, x = 3 and the x − axis is (a) 4

(b) 2

(c) 3

(d) 1.

185. The area bounded by the curves y = log e x and y = ( log e x ) is 2

(a) 3 − e

(b) e − 3

(c)

1 (3 − e ) 2

(d)

1 (e − 3) . 2

186. The area of figure bounded by y = e x , y = e − x and the straight line x = 1 is (a) e +

1 e

(b) e −

1 e

(c) e +

1 −2 e

1 (d) e + + 2 . e

187. The area bounded by the curves y = x , 2 y + 3 = x and x − axis in the 1st quadrant is (a) 9

(b)

27 4

(c) 36

(d) 18 .

188. The area enclosed between the curve y = log e ( x + e ) and the co - ordinate axes is (a) 3

(b) 4

(c) 1

(d) 2 .

189. The parabolas y 2 = 4 x and x 2 = 4 y divide the square region bounded by the lines x = 4 , y = 4 and the coordinate axes. If S1 , S 2 , S3 are respectively the areas of these of these parts numbered from top to bottom , then S1 : S2 : S3 is (a) 2 : 1 : 2

(b) 1 : 1 : 1

(c) 1: 2 : 1

(d) 1 : 2 : 3.

190. If A is the area of the region bounded by the curve y = 3 x + 4, x axis and the line x = −1 and B is that area bounded by curve y 2 = 3 x + 4, x = axis and the lines x = −1 and x = 4 then A : B is equal to (a) 1 : 1

(b) 2 : 1

(c) 1: 2

(d) None of these.

191. The area bounded by the curve y = ( x + 1) , y = ( x − 1) and then line y = 2

(a) 1/6

(b) 2/3

2

(c) 1/4

1 is 4

(d) 1/3.

192. Let f ( x ) be a non - negative continous function such that the area bounded by the curve y = f ( x ) , x − axis and the ordinates x =  

(a)  1 −

π  − 2 4 

 

(b)  1 −

π π π π    , x = β > is  β sin β + cos β + 2 β  then f   is 4 4 4   2 π  + 2 4 

π (c)  + 2 − 1 4



π (d)  − 2 + 1 . 4



193. Let y be the function which passes through ( 1, 2 ) having slope ( 2x + 1) . The area bounded between the curve and x = axis is (a) 6 sq. unit

(b) 5 / 6 sq. unit

(c) 1/ 6 sq. unit

(d) None of these.

L E V E L - 3 (Tougher Problems) 1. Let f ( x ) be a function satisfying f ′ ( x) = f ( x ) with f (0 ) = 1 and g ( x ) be the function satisfying

f ( x ) + g ( x ) = x . The value of integral 2

1

∫ f ( x )g ( x ) dx is equal to 0

1 (a) ( e − 7 ) 4

1 (b) (e − 2 ) 4

(c)

1 (e − 3 ) 2

(d) None of these.

k

k

1−k

1−k

2. Let f be a positive function . Let I1 = ∫ xf (x(1− x)) dx, I2 = ∫ f (x(1− x)) dx when 2k − 1 > 0. Then I1 / I 2 is (a) 2 3.

∫

1

0

x7 1 − x4

(b) k

(c)1/2

(d) 1.

dx is equal to

(a) 1

(b)

4. If n is any integer, then (a) x

∫

π

0

1 3

(c)

2 3

(d)

π . 3

ecos x cos3 ( 2n + 1) xdx = 2

(b) 1

(c) 0

(

(d) None of these.

)(

)

(

)(

)

5. Let a, b, c be non-zero real numbers such that ∫ 1 + cos8 x ax 2 + bx + c dx = ∫ 1 + cos8 x ax 2 + bx + c dx . 1

0

2

0

Then the quadratic equation ax 2 + bx + c = 0 has (a) No root in (0, 2)

(b) At least one root in (0, 2)

(c) A double root in (0, 2)

(d) None of these .

6. If f ( x ) = ∫ t dt , x ≥ −1, then x

−1

(a) f and f ' are continous for x + 1 > 0

(b) f is continous but f ′ is not continous for x + 1 > 0

(c) f and f ′ are not continous at x = 0

(d) f is continous at x = 0 but f ′ is not so.

1 1 ≤ f (t ) ≤ 1, t ∈ [0,1] and 0 ≤ f (t ) ≤ for t ∈ (1, 2], then 2 2 3 5 (b) 0 ≤ g ( 2) < 2 (c) < g ( 2 ) ≤ (d) 2 < g ( 2 ) < 4 . 2 2

7. Let g ( x ) = ∫ f (t ) dt where x

0

3 1 (a) − ≤ g ( 2 ) < 2 2

8. The value of (a) π

cos 2 x ∫−π 1 + a x dx, a > 0, is π

(b) aπ

π (c) 2

(d) 2π .

9. If f ( x ) =

f (a ) ex I f (a ) xg ( x (1 − x )) dx, and I 2 = ∫ , I1 = ∫ g ( x (1 − x )) dx, then the value of 2 is x f a − ( ) − f ( a) 1+ e I1

(b) −3

(a) 1

∫ f ( x ) dx = 1, ∫

x f ( x ) dx = a, and 0

1

10. Let

0

1

(c) −1

(d) 2.

x 2 f ( x ) dx = a 2 , then the value of 0

∫

1

∫ ( x − a ) f ( x ) dx = 1

2

0

(b) a 2 (c) a 2 − 1 (d) a 2 − 2a + 2 . ∞ ∞ x 2 dx π x 2 dx = 11. Given that ∫0 2 then the value of ∫0 ( x 2 + 4 )( x 2 + 9 ) is ( x + a 2 )( x 2 + b2 )( x 2 + c 2 ) 2 (a + b )(b + c )(c + a ) (a) 0

π 60

(a)

(b)

π 20

π 40

(c)

(d)

π . 80

12. If l ( m, n ) = ∫0 t m (1 + t ) dt , then the expression for l ( m, n) in terms of l ( m + 1, n − 1) is 1

n

2n n l ( m + 1, n − 1) − m +1 m +1

(b)

n l ( m + 1, n − 1) m +1

2n n l ( m + 1, n − 1) + (c) m +1 m +1

(d)

m l ( m + 1, n − 1) n +1

(a)

4 4 4 3 3 3 13. lim 1 + 2 + 3 + ..... + n − lim 1 + 2 + 3 + .... + n = n →∞ n →∞ n5 n5 1 1 (b) Zero (c) (a) 30 4

∫

1 . 5

t2

2 xf ( x ) dx = t 5 , t > 0, then f  4  = 5  25  2 5 (a) (b) 5 2

14. If

(d)

0

2 (d) None of these. 5 15. For which of the following values of ‘ m ’, the area of the region bounded by the curve y = x − x 2 and the 9 line y = mx equals 2 (a) - 4 (b) - 2 (c) 2 (d) 4.

(c) −

16. Area enclosed between the curve y 2 ( 2a − x ) = x 3 and line x = 2a above x − axis is (a) π a

2

3π a 2 (b) 2

(c) 2π a 2

(d) 3π a 2 .

17. What is the area bounded by the curves x 2 + y 2 = 9 and y 2 = 8 x is 2 2 9π 1 + − 9sin −1   3 2 3

(a) 0

(b)

(c) 16π

(d) None of these.

18. The area bounded by the curves y = x − 1 and y = − x + 1 is (a) 1

(b) 2

(c) 2 2

(d) 4.

19. If for a real number y, [ y ] is the greatest integer less than or equal to y , then the value of the integral 3π / 2

∫ [2sin x ] dx is

π /2

(a) −π

(c) −

(b) 0

πx 1 20. If f ( x ) = A sin   + B, f '   = 2 and  2  2 π π 2 3 (a) and (b) and 2 2 π π

21. I n = ∫

π /4

0

π 2

(d)

π . 2

2A ∫ f ( x ) dx = π , then the constants A and B are respectively 1

0

4 and 0 π

(c)

(d) 0 and −

4 . π

[ I n + I n− 2 ] equals tan n x dx, then lim n →∞

(a) 1/ 2

(c) ∞

(b) 1

(d) 0.

22. The area bounded by the curves y = In x, y = In x , y = in x and y = in x is (b) 6 sq. unit

(a) 4 sq. unit

(c) 10 sq. unit

(d) None of these.

(c) π

(d) 0.

(b) α < 0

(c) 0 < α < 1

(d) None of these.

(b) 8

(c) 10

(d) 18.

(b) π 2

(c) 0

(d) π / 2 .

1  sin  n +  x 2 23. ∫  dx, ( n ∈ N ) equals sin x 0 π

(b) ( 2n + 1)

(a) nπ

∫

24. If

1

0

e x ( x − α ) dx = 0, then 2

(a) 1 < α < 2 25.

∫

10π

π

sin x dx is

(a) 20 26.

∫

π

−π

π 2

2 x (1 + sin x ) dx is 1 + cos 2 x

(a) π 2 / 4

27. If I1 = ∫ 2 x dx, I 2 = ∫ 2 x3dx, I 3 = ∫ 2 x dx, I 4 = ∫ 2 x dx, then 1

1

2

0

2

0

(a) I 3 = I 4

1

(b) I 3 > I 4 1

28. If 2 f ( x ) − 3 f   = x, then x (a)

3 In 2 5

(b)

2

2

3

1

(c) I 2 > I1

(d) I1 > I 2 .

∫ f ( x ) dx is equal to 2

1

−3 (1 + In 2 ) 5

(c)

−3 In 2 5

(d) None of these.

ANSWER KEY LEVEL - 1 (Fundamentals of Definite Integration) 1.

c

24.

b

47.

b

70.

b

2.

a

25.

a

48.

c

71.

b

3.

c

26.

c

49.

b

72.

a

4.

b

27.

c

50.

a

73.

d

5.

c

28.

b

51.

d

74.

c

6.

a

29.

c

52.

a

75.

a

7.

b

30.

d

53.

a

76.

c

8.

d

31.

d

54.

a

77.

b

9.

a

32.

a

55.

b

78.

a

10.

c

33.

b

56.

c

79.

d

11.

d

34.

b

57.

b

80.

b

12.

c

35.

d

58.

c

81.

b

13.

a

36.

b

59.

b

82.

c

14.

d

37.

a

60.

d

83.

d

15.

a

38.

a

61.

b

84.

a

16.

b

39.

c

62.

a

85.

a

17.

c

40.

c

63.

d

86.

b

18.

d

41.

b

64.

c

87.

b

19.

d

42.

b

65.

a

88.

a

20.

c

43.

a

66.

c

89.

a

21.

b

44.

b

67.

a

90.

b

22.

a

45.

a

68.

d

91.

c

23.

c

46.

c

69.

d

92.

c

ANSWER KEY LEVEL - 2 (Properties of Definite Integration) 1.

b

29. d

57. c

85. c

113. a

141. c

169. a

2.

c

30. a

58. a

86. c

114. c

142. a

170. c

3.

d

31. a

59. b

87. c

115. b

143. a

171. b

4.

b

32. b

60. c

88. a

116. b

144. d

172. d

5.

b

33. b

61. a

89. b

117. a

145. a

173. b

6.

d

34. a

62. b

90. c

118. b

146. c

174. c

7.

a

35. c

63. a

91. c

119. a

147. a

175. c

8.

c

36. c

64. b

92. a

120. a

148. c

176. d

9.

d

37. d

65. c

93. d

121. d

149. d

177. a

10. c

38. a

66. b

94. d

122. d

150. b

178. c

11. d

39. b

67. a

95. d

123. b

151. b

179. b

12. c

40. a

68. b

96. b

124. b

152. c

180. b

13. d

41. b

69. d

97. c

125. b

153. c

181. a

14. a

42. a

70. d

98. c

126. b

154. b

182. d

15. a

43. d

71. c

99. d

127. b

155. d

183. b

16. d

44. d

72. a

100. a

128. d

156. b

184. d

17. b

45. c

73. b

101. b

129. a

157. d

185. a

18. c

46. b

74. c

102. c

130. b

158. a

186. c

19. b

47. c

75. a

103. c

131. a

159. b

187. a

20. b

48. d

76. c

104. c

132. b

160. d

188. c

21. c

49. d

77. a

105. c

133. d

161. b

189. b

22. d

50. c

78. a

106. c

134. a

162. b

190. a

23. b

51. a

79. d

107. d

135. b

163. a

191. d

24. d

52. b

80. b

108. a

136. c

164. c

192. b

25. b

53. b

81. c

109. d

137. b

165. b

193. c

26. c

54. a

82. b

110. a

138. a

166. d

27. b

55. c

83. a

111. b

139. d

167. a

28. b

56. b

84. c

112. a

140. c

168. b

ANSWER KEY LEVEL - 3 (Tougher Problems)

1.

d

8.

c

15. b

22. a

2.

c

9.

d

16. b

23. c

3.

b

10. a

17. b

24. c

4.

c

11. a

18. b

25. d

5.

b

12. a

19. c

26. b

6.

a

13. d

20. c

27. d

7.

b

14. a

21. b

28. b