definite integration assignment for iit-jee

7/29/2019 Definite INTEGRATION ASSIGNMENT FOR IIT-JEE

1/36

LEVEL - 1 (Fundamentals of Definite Integration)

1.1

21

0

nxe dx =

(a) 0 (b)1

2(c)

1

3(d)

1

4.

2./ 4

2

0tan xdx

=

(a) 14 (b) 1 4+ (c) 14

(d) 4

.

3./ 2

0

sin

1 cos

x xdx

x

+=

+

(a) log 2e (b) log 2e (c) 2

(d) 0.

4./ 2

0sinxe xdx

=

(a) ( )/ 21 1

2e (b) ( )

/ 21 12

e + (c) ( )/ 21 1

2e (d) ( )

/ 22 1e + .

5.2

21

1 1xe dx

x x

=

(a)

2

2

ee+ (b)

2

2

ee (c)

2

2

ee (d) None of these.

6. ( )( )

/ 2

0

cos

1 sin 2 sin

xdx

x x

=+ +

(a)4

log3

(b)1

log3

(c)3

log4

(d) None of these.

7.

( )

/ 2

5/ 32

1 cos

1 cos

xdx

x

+=

(a)5

2(b)

3

2(c)

1

2(d)

2

5.

8.

12

21

1xe dx

x

=

(a) 1e + (b) 1e (c)1e

e

+(d)

1e

e

.


2/36

9.1

1

20

2sin

1

xdx

x

= +

(a) 2 log 22

(b) 2 log 2

2

+ (c) log 2

4

(d) log 2

4

+ .

10. The value of ( )2

32 ax bx c + + depends on

(a) The value of a (b) The value of b (c) The value of c (d) The value of a and b.

11./ 4

/ 6cos 2ec x dx

=

(a) log3 (b) log 3 (c) log9 (d) None of these.

12.logb

a

xdx

x=

(a)log

loglog

b

a

(b) ( )log logb

a ba

(c) ( )1

log log2

ba b

a

(d) ( )1

log log2

aa b

b

13.1

1

0tan x dx

=

(a)1

log24 2

(b)

1log2

2 (c) log2

4

(d) log2 .

14.( )( )

1

201

dx

ax b x=

+

(a)a

b(b)

b

a(c) a b (d)

1

ab.

15./ 2

2

/ 4cos cos ec d

=

(a) 2 1 (b) 1 2 (c) 2 1+ (d) None of these .

16. ( )

11/ 2

3/ 20 2

sin

1

xdx

x

=

(a)1

log 24 2

e

+ (b)

1log 2

4 2e

(c) log 2

2e

+ (d) log 2

2e

.

17./ 2

0 2 cos

dx

x

=+

(a) 11 1tan3 3

(b) ( )13 tan 3 (c) 12 1tan3 3

(d) ( )12 3 tan 3 .


3/36

18.

11

20

tan

1

xdx

x

=+

(a)

2

8

(b)

2

16

(c)

2

4

(d)

2

32

.

19. The value of integral( )2 /

21/

sin 1/xdx

x

=

(a) 2 (b) -1 (c) 0 (d) 1.

20.2

/ 2

0.sin

2 4

x xe dx + =

(a) 1 (b) 2 2 (c) 0 (d) None of these.

21.1

0 1

x

xe dxe

=+

(a)1 1

log 1e

e e

+ +

(b)1 1

log 12

e

e e

+ +

(c)1 1

log 12

e

e e

+ +

(d) None of these.

22./ 4

0

sin cos

9 16 sin 2

x xdx

x

+=

+

(a)1

log320

(b) log3 (c)1

log520

(d) None of these.

23. ( )/ 2

/ 4logsin cot

xe x x dx

+ =

(a) / 4 log2e (b) / 4 log2e

(c)/ 41

log22

e

(d) / 41

log22

e .

24.

11/ 2

20

sin

1

x xdx

x

=

(a)1 3

2 12

+ (b)

1 3

2 12

(c)

1 3

2 12

(d) None of these.

25.2

0

2

2

xdx

x

+=

(a) 2+ (b)3

2 + (c) 1+ (d) None of these.

26.0 1 sin

dx

x

=+

(a) 0 (b)

1

2 (c) 2 (d)

3

2 .


4/36

27.2

01 sin

2

xdx

+ =

(a) 0 (b) 2 (c) 8 (d) 4.

28.1

1

0cos x dx =

(a) 0 (b) 1 (c) 2 (d) None of these.

29./ 2

0

cos

1 cos sin

xdx

x x

=+ +

(a)1

log24 2

+ (b) log2

4

+ (c)

1log2

4 2

(d) log2

4

.

30. ( )/ 6

2

02 3 cos 3x xdx

+ =

(a) ( )1

1636

+ (b) ( )1

1636

(c) ( )21

1636

(d) ( )21

1636

+ .

31./ 2

40

sin cos

1 sin

x xdx

x

=+

(a)2

(b)

4

(c)

6

(d)

8

.

32./ 4

6 2

0tan secx x dx

=

(a)1

7(b)

2

7(c) 1 (d) None of these.

33./ 6

30

sin

cos

xdx

x

=

(a)

2

3 (b)

1

6 (c) 2 (d)

1

3 .

34./ 2

20

sin cos

cos 3cos 2

x x dxdx

x x

=+ +

(a)8

log9

(b)9

log8

(c) ( )log 8 9 (d) None of these.

35. The value of the definite integral1

200

2 cos 1

dxfor

x x

< (c) 1 2I I< (d) None of these.

38./ 2

/ 4sinxe xdx

=

(a) / 21

2e

(b) / 42

2e

(c) ( )/ 4 / 42 e e + (d) 0.

39.( )

/2

20

1 2cos

2 cos

xdx

x

+ =+

(a)2

(b) (c)

1

2(d) None of these.

40. 20 1 2 cos

dx

a x a

= +

(a)

( )2

2 1 a

(b) ( )21 a (c) 21 a

(d) None of these.

41. ( )1 9

01 x dx =

(a) 1 (b)1

10(c)

11

10(d) 2.

42./ 3

0cos3xdx

=

(a) (b) 0 (c) 2

(d) 4

.

43. The value of/ 4

0

1 tan

1 tan

xdx

x

+ is

(a)1

log22

(b)1

log24

(c)1

log23

(d) None of these.

44. The value of1

0 x x

dx

e e+ is

(a) 11

tan1

e

e

+ (b)

1 1tan

1

e

e

+ (c)

4 (d) 1tan

4e + .


6/36

45.1

1 loge xdx

x

+=

(a)3

2(b)

1

2(c)

1

e(d) None of these.

46. If

1

0

2

log 1 log ,2 3

x

x dx a b

+ = + then

(a)3 3

,2 2

a b= = (b)3 3

,4 4

a b= = (c)3 3

,4 2

a b= = (d) a b= .

47.1

0 1

dx

x x=

+

(a)2 2

3(b)

4 2

3(c)

8 2

3(d) None of these.

48./ 4

4 40

4sin2sin cos

d

=+

(a) / 4 (b) / 2 (c) (d) None of these.

49.( )

( )

1

30

1

1

xe xdx

x

=

+

(a)4

e(b) 1

4

e (c) 1

4

e+ (d) None of these.

50. If ( ) ( )4 1 1,x x x+ = then ( )2

1x dx =

(a)1 32

log4 17

(b)1 32

log2 17

(c)1 16

log4 17

(d) None of these.

51.1/ 2

21/ 4

dx

x x=

(a) (b)2

(c)

3

(d)

6

.

52. The value of2

0

3 xdx

x is

(a) ( )22

3 1log3

(b) 0 (c)2 2

log 3(d)

23

2.

53. ( )2

0sin cosx x dx

+ =

(a) 0 (b) 2 (c) -2 (d) 1.


7/36

54./ 4

20

sec

1 2 sin

x

x

+ is equal to

(a) ( )1

log 2 13 2 2

+ + (b) ( )

1log 2 1

3 2 2

+

(c)

( )3 log 2 1

2 2

+

(d)

( )3 log 2 1

2 2

+ +

55. The value of/ 2

20

sin

1 cos

xdx

x

+ is

(a) / 2 (b) / 4 (c) / 3 (d) / 6 .

56. The value of2

1logx dx is

(a) log2/e (b) log4 (c) log4/e (d) log2 .

57. The value of

25

23 4

xdx

x is

(a)15

2 log7

e

(b)15

2 log7

e

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

e e e+ + (d) 1 152 tan

7

.

58. The value of2 2sin cos

1 1

0 0sin cos

x x

tdt tdt +

(a)

2

(b) 1 (c)

4

(d) None of these.

59. If for non-zero x , ( )1 1

5,af x bf x x

+ =

where ,a b then ( )2

1f x dx =

(a)( )2 2

1 7log 2 5

2a a b

a b

+ + (b)

( )2 21 7

log 2 52

a a ba b

+

(c)( )2 2

1 7log 2 5

2a a b

a b

(d)

( )2 21 7

log 2 52

a a ba b

+ .

60. If / 4

0

tan ,n

nI d

=

then 8 6I I+ equals

(a)1

4(b)

1

5(c)

1

6(d)

1

7.

61.2 / 3

20 4 9

dx

x=

+

(a)12

(b)

24

(c)

4

(d) 0.

62. The value of

41

20

1

1

xdx

x

+

+is

(a) ( )1

3 46

(b) ( )1

3 46

(c) ( )1

3 46

+ (d) ( )1

3 46

+ .


8/36

63.2 3

0sin

a

x x dx equals

(a) ( )31 cos a (b) ( )33 1 cos a (c) ( )31

1 cos3

a (d) ( )31

1 cos3

a

64. ( )/ 4

0tan cotx x dx

+

equals

(a) 2 (b)2

(c)

2

(d) 2 .

65.1

0

1

1

xdx

x

+

equals

(a) 12

(b) 12

+

(c)2

(d) ( )1+ .

66.1

1edx

x

is equal to

(a) (b) 0 (c) 1 (d) ( )log 1 e+ .

67.

2

1

logx xdx

x=

(a) ( )2

log x (b) ( )21

log2

x (c)

2log

2

x(d) None of these.

68./ 2

2 2 2 20

cos sin

dx

a x b x

=

+

(a) ab (b)2ab (c)

ab

(d)

2ab

.

69. ( ) ( ) ( )/ 4 5 / 4 / 4

0 / 4 2cos sin sin cos cos sinx x dx x x dx x x dx

+ + is equal to

(a) 2 2 (b) 2 2 2 (c) 3 2 2 (d) 4 2 2 .

70. ( ) ( )0 4 ,a

x dx a + then

(a) 0 4a (b) 2 4a (c) 2 0a (d) 2 4a or a .

71.0

21 2 2

dx

x x=

+ +

(a) 0 (b) / 4 (c) / 2 (d) / 4 .

72.3

21

1

1

dx

x+ is equal to

(a) /12 (b) / 6 (c) / 4 (d) / 3 .


9/36

73.3

1( 1)( 2)( 3)x x x dx =

(a) 3 (b) 2 (c) 1 (d) 0.

74.3

22

dx

x x

=

(a) ( )log 2/3 (b) ( )log 1/4 (c) ( )log 4/3 (d) ( )log 8/3 .

75. ( )

15

8 3 1

dx

x x=

+

(a)1 5

log2 3

(b)1 5

log3 3

(c)1 3

log2 5

(d)1 3

log5 5

.

76. The value of3

0sin d

is

(a) 0 (b) 3 / 8 (c) 4 / 3 (d) .

77.1

1

0

1sin 2 tan

1

xdx

x

+ =

(a) / 6 (b) / 4 (c) / 2 (d) .

78.3

20

3 1

9

xdx

x

+=

+

(a) ( )log 2 212

+ (b) ( )log 2 2

2

+ (c) ( )log 2 2

6

+ (d) ( )log 2 2

3

+

79. The value of( )

2

41 1

dx

x x+ is

(a)1 17

log4 32

(b)1 17

log4 2

(c)17

log2

(d)1 32

log4 17

.

80. The value of( )

3

22

1

1

xdx

x x

+ is

(a)1

2 log 26

(b)16 1

log9 6

(c) 4 1log3 6

(d)16 1

log9 6

+ .

81. The value of1

loge

x dx is

(a) 0 (b) 1 (c) 1e (d) 1e + .

82. The value of( )

2/ 2

0

sin cos

1 sin 2

x xI dx

x

+=

+is

(a) 3 (b) 1 (c) 2 (d) 0.


10/36

83./ 8

3

0co s 4 d

=

(a)2

3(b)

1

4(c)

1

3(d)

1

6.

84. ( )

8

3

2 3

1

x

dxx x

+ is equal to

(a) ( )32log 3/ 2e (b) ( )3log 3/ e (c) ( )34log 3/e (d) None of these

85. The value of1

2

0

xx e dx is equal to

(a) 2e (b) 2e + (c) 2 2e (d) 2e .

86. Let

2 2

1 221 11

dx dx

I and I xx= =+ then

(a)1 2I I> (b) 2 1I I> (c) 1 2I I= (d) 1 221I > .

87. The value of( )

tan cot

2 21/ /1 1

x x

e I e

t dt dt

t t t+ =

+ +

(a) -1 (b) 1 (c) 0 (d) None of these.

88.

3 / 4

/ 4 1 cos

dx

x

+ is equal to

(a) 2 (b) -2 (c)1

2(d)

1

2 .

89. The value of( )

2

21 1

e dx

x ln x+is

(a) 2 / 3 (b) 1/ 3 (c) 3 / 2 (d) ln 2.

90.

/ 22

/ 4 cosec xdx

=(a) -1 (b) 1 (c) 0 (d)

1

2.

91. If( )

1/ 2log2,

61

x

u

du

e

=

then xe =

(a) 1 (b) 2 (c) 4 (d) -1.

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

2 1

1 ' ( ) 'fg x f g x g x dx

is equal to(a) 1 (b) 2 (c) 0 (d) None of these.


11/36

LEVEL - 2 (Properties of Definite Integration)

1. ( )0

sinxf x dx

=

(a) ( )0

sinf x dx

(b) ( )0 sin2 f x dx

(c) ( )/ 2

0sin

2f x dx

(d) None of these.

2./ 2

0

cot

cot tan

xdx

x x

=+

(a) (b)2

(c)

4

(d)

3

.

3./ 2

0 1 tan

d

=

+

(a) (b)2

(c)

3

(d)

4

.

4. If ( ) 3 ,x

t

af x t e dt= then ( )

df x

dx=

(a) ( )3 23xe x x+ (b) 3 xx e (c) 3 aa e (d) None of these.

5.1

1x x dx

=

(a) 1 (b) 0 (c) 2 (d) -2.

6./ 2

0log tan x dx

=

(a) log 22

e

(b) log 2

2e

(c) log 2

e (d) 0.

7./ 2

0logsin x dx

=

(a) log22

(b)1

log2

(c)1

log2

(d) log22

.

8./ 2

0

cos sin

1 sin cos

x xdx

x x

=

+

(a) 2 (b) -2 (c) 0 (d) None of these.


12/36

9.1

1

2log

2

xdx

x

= +

(a) -2 (b)1 (c) -1 (d) 0.

10.1

17 4

1

cosx xdx

=

(a) -2 (b)-1 (c) 0 (d) 2.

11.

3/ 2/ 2

3/ 2 3/ 20

sin

cos sin

xdx

x x

=+

(a) 0 (b) (c) / 2 (d) / 4 .

12. ( )/ 4

0

log 1 tan d

+ =

(a) log2

4

(b)

1log

4 2

(c) log2

8

(d)

1log

8 2

.

13.2

0

sin 2

cosd

a b

=

(a) 1 (b) 2 (c)4

(d) 0.

14. ( )1

01f x dx

has the same value as the integral

(a) ( )1

0f x dx (b) ( )

1

0f x dx (c) ( )

1

01f x dx (d) ( )

1

1f x dx

.

15. ( )1/ 2

1/ 2

1cos log

1

xx dx

x

= +

(a) 0 (b) 1 (c) 1/ 2e (d)1/ 22e .

16. The value of1

20 1

dx

x x+ is

(a)3

(b)

2

(c)

1

2(d)

4

.

17. If ( )1

10,f x

= then

(a) ( ) ( )f x f x= (b) ( ) ( )f x f x = (c) ( ) ( )2f x f x= (d) None of these.

18.1

11 x dx

=

(a) -2 (b) 0 (c) 2 (d) 4.


13/36

19.3

0sinx x dx

=

(a)4

3

(b)

2

3

(c) 0 (d) None of these.

20.2

2

2

1 x dx

=

(a) 2 (b) 4 (c) 6 (d) 8.

21./ 2

0

cos

sin cos

xdx

x x

=+

(a) 2 (b)2

(c)

4

(d) None of these.

22./ 2

4 40

sin cos

cos sin

x x xdx

x x

=+

(a) 0 (b)8

(c)

2

8

(d)

2

16

.

23. The correct evaluation of/ 2

0sin

4x dx

is

(a) 2 2+ (b) 2 2 (c) 2 2 + (d) 0.

24. ( )0

a

f x dx =

(a) ( )0

a

f a x dx+ (b) ( )0 2a

f a x dx+ (c) ( )0a

f x a dx (d) ( )0a

f a x dx .

25./ 2

0sin cosx x dx

=

(a) 0 (b) ( )2 2 1 (c) 2 1 (d) ( )2 2 1+ .

26.0

cosx dx

=

(a) (b) 0 (c) 2 (d) 1.

27. The value of the integral/ 4

4

/ 4sin x dx

=

(a) 3/2 (b) -8/3 (c) 3/8 (d) 8/3.

28.1.5

2

0,x dx where [ . ] denotes the greatest integer function, equals

(a) 2 2+ (b) 2 2 (c) 2 2 + (d) 2 2 .


14/36

29.0

tan

sec tan

x xdx

x x

=+

(a) 12

(b) 1

2

+

(c) 12

+ (d) 1

2

.

30. 0

tan

sec cos

x x

dxx x

=+(a)

2

4

(b)

2

2

(c)

23

2

(d)

2

3

.

31.1

3 2

1sin cosx xdx

=

(a) 0 (b) 1 (c)1

2(d) 2.

32. For any integer n , the integral ( )

2sin 3

0 cos 2 1

x

e n x dx

+ =(a) -1 (b) 0 (c) 1 (d) .

33.1/

loge

ex dx =

(a)1

1e

(b)1

2 1e

(c) 1 1e (d) None of these.

34. [ ]( )/ 2

0

sinx x dx

is equal to (where [.] represents greatest integer function)

(a)

2

8

(b)

2

18

(c)

2

28

(d) None of these.

35. The value of the integral ( )1

01

n

I x x dx= is

(a)1

1n +(b)

1

2n +(c)

1 1

1 2n n

+ +(d)

1 1

1 2n n+

+ +.

36. The value of [ ]2

2sin ,x dx

where [ . ] represents the greatest integer function, is

(a) (b) 2 (c)5

3

(d)

5

3

.

37. The value of/ 2

30 1 tan

dx

x

+ is

(a) 0 (b) 1 (c)2

(d)

4

.

38. The value of3 / 4

/ 4,

1 sind

+ is

(a) tan8

(b) log tan

8

(c) tan

8

(d) None of these.


15/36

39. If ( ) ( )( ),b

af a b x f x then x f x dx+ = =

(a) ( )2

b

a

a bf b x dx

+ (b) ( )2

b

a

a bf x dx

+ (c) ( )2

b

a

b af x dx

(d) None of these.

40.0

sinx x dx

=

(a) (b) 0 (c) 1 (d) 2 .

41. If ( ) ( )2

0 02 ,

a a

f x dx f x dx= then

(a) ( ) ( )2f a x f x = (b) ( ) ( )2f a x f x = (c) ( ) ( )f a x f x = (d) ( ) ( )f a x f x = .

42. If / 4 / 4

2 2

0 0sin cos ,I x dx and J x dx then I

= = =

(a)4

J

(b) 2J (c) J (d)2

J.


13 1x x dx + is

(a) 10 (b)5

6(c) 21 (d) 12.

44. The value of3

2 5

xdx

x x + is

(a) 1 (b) 0 (c) -1 (d)1

2.

45. The value of2cos 5

0cos 3xe x dx

is

(a) 1 (b) -1 (c) 0 (d) None of these.

46./ 2

0

1

1 tandx

x

=+

(a)2

(b)

4

(c)

6

(d) 1.

47. The value of2

1

1

sin

3

x xdx

x

is

(a) 0 (b)1

0

sin2

3

xdx

x (c)2

1

02

3

xdx

x

(d)

21

0

sin2

3

x xdx

x

.

48.1

11

1sin xdx

is equal to

(a)1 0 8 6 4 2

. . . .11 9 7 5 3

(b)1 0 8 6 4 2

. . . . .11 9 7 5 3 2

(c) 1 (d) 0.


16/36

49. To find the numerical value of ( )2

2

2,px qx s dx

+ + it is necessary to know the values of constants

(a) p (b) q (c) s (d) p and s.

50.1

1

1log

1

xdx

x

+ =

(a) 2 (b) 1 (c) 0 (d) .

51./ 2

/ 2

cos

1 xx

dxe

=

+

(a) 1 (b) 0 (c) -1 (d) None of these.

52. If [ ]x denotes the greatest integer less than or equal to x , then the value of the integral [ ]2

2

0

x x dx

equals

(a) 5/3 (b) 7/3 (c) 8/3 (d) 4/3.

53.3

0cos x dx

=

(a) -1 (b) 0 (c) 1 (d) .

54.2

0logsin x dx

=

(a)

12 log

2e

(b) lo g 2e

(c)1

log2 2

e

(d) None of these.

55. If ( )f x is an odd function ofx , then ( )22

cosf x dx

is equal to

(a) 0 (b) ( )20

cosf x dx

(c) ( )202 sinf x dx

(d) ( )0 cosf x dx

.

56.2

0sin x dx

is equal to

(a) (b)2


57./ 2

0

sin

sin cos

xdx

x x

+ equals

(a)2

(b)

3

(c)

4

(d)

6

.

58.1

1

1tanx x dx

equals

(a) 12

(b) 12

+

(c) ( )1 (d) 0.


17/36

59. ( )sin cosa

ax f x dx

=

(a) ( )0

2 sin cosa

xf x dx (b) 0 (c) 1 (d) None of these.

60. The value of2

3

0sin d

is

(a) 0 (b) 3 / 8 (c) 8 / 3 (d) .

61.2

1x dx

(a) 5 / 2 (b) 1 / 2 (c) 3 / 2 (d) 7 / 2.

62.3

02 x dx equals

(a) 2 / 7 (b) 5 / 2 (c) 3 / 2 (d) -3 / 2.

63. The value of

sin/ 2

sin cos0

2

2 2

x

x xdx

+ is

(a)4

(b)

2

(c) (d) 2 .

64. The value of1

2

03 1x dx is

(a) 0 (b) 4 / 3 3 (c) 3 / 7 (d) 5 / 6.

65.2/ 2 cos

2/ 2

sin

1 cos

xxe dx

x

+ is equal to

(a) 12e (b) 1 (c) 0 (d) None of these.

66. ( ) ( )2 ,f x f x= then ( )1.5

0.5xf x dx equals

(a) ( )1

0f x dx (b) ( )

1.5

0.5f x dx (c) ( )

1.5

0.52 f x dx (d) 0.

67.

2

2

2

/ 2

0

2

x

xx

ed x

e e

+

is

(a) / 4 (b) / 2 (c)2

/16e (d)

2

/ 4e .

68. If [ ]x denotes the greatest integer less than or equal to x , then the value of5

1

3x dx

is

(a) 1 (b) 2 (c) 4 (d) 8.


18/36

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 ( ) ( )1 0

0 15,f x dx then f t dt

= =

(a) 10 (b) 5 (c) 0 (d) -5.

71. Let ( ) ( )1 2sin , sin ,a a

a aI xf x dx I f x dx

= = then 2I is equal to

(a) 12

I

(b)1I (c) 1

2I

(d)

12I .

72.1/ 2

1/ 2

1cos .

1

xx ln dx

x

+ is equal to

(a) 0 (b) 1 (c) 2 (d) ln 3.

73. The value of2

1

logee

e

xdx

x is

(a)3

2(b)

5

2(c) 3 (d) 5.

74. If ( ) ( )3

2

sin , 2,

2,

cosxe x x

f x then f x dxotherwise

=

is equal to

(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

(a) 0 (b) (c) 1 (d) None of these.

76./ 3

/ 6

1 cot

dx

x

+is

(a) / 3 (b) / 6 (c) /12 (d) / 2 .

77. The value of

2/ 3/ 2

2 / 3 2 / 30

sin

sin cos

xdx

x x

+ is

(a) / 4 (b) / 2 (c) 3 / 4 (d) .

78. ( )1

2

1log 1x x dx

+ + =

(a) 0 (b) log2 (c)1

log2

(d) None of these.


19/36

79. The value of the integral ( ) ( )2

cos sin , intax bx dx a and b are eger

is

(a) (b) 0 (c) (d) 2 .

80.0

1 cos 2

2

xdx

+

is equal to

(a) 0 (b) 2 (c) 1 (d) -1.

81. ( )2

0

a

f x dx =

(a) ( )0

2a

f x dx (b) 0 (c) ( ) ( )0 0 2a a

f x dx f a x dx+ (d) ( ) ( )2

0 02

a a

f x dx f a x dx+ .

82.2sin 3

0cos

xe x dx

is equal to

(a) -1 (b) 0 (c) 1 (d) .

83. Find the value of9

02 ,x dx + where [ . ] is the greatest integer function

(a) 31 (b) 22 (c) 23 (d) None of these.

84. The value of2

2

0,x dx where [ . ] is the greatest integer function

(a) 2 2 (b) 2 2+ (c) 2 1 (d) 2 2 .

85.[ ]1000

0

x xe dx

is

(a) 1000 1e (b)1000

1

1

e

e

(c) ( )1000 1e (d)1

1000

e .

86. The value of the ingral

1

1

an

n

n

xdx

a x x

+ is

(a)2a (b) 2

2na

n+ (c) 2

2na

n (d) None of these.

87./ 2

0sin2 logtanx x dx

is equal to

(a) (b) / 2 (c) 0 (d) 2 .

88. The integral [ ]1/ 2

1/ 2

1log

1

xx dx

x

+ + equal ( where [.] is the greatest integer function )

(a) 12

(b) 0 (c) 1 (d) 12log2

.


20/36

89. ( )2

0sin sinx x dx

+ =

(a) 0 (b) 4 (c) 8 (d) 1.

90. The value of ( )/ 2

3

/ 23sin sinx x dx

+ is

(a) 3 (b) 2 (c) 0 (d)10

3.

91. The value of1

0

1

2I x x dx= is

(a) 1/3 (b) 1/4 (c) 1/8 (d) None of these.

92. The value of8

05x dx

(a) 17 (b) 12 (c) 9 (d) 18.

93.2

01x dx =

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

94. [ ]2

2x dx

= (where [.] denotes greatest integer function)

(a) 1 (b) 2 (c) 3 (d) 4.

95.1

1

20

1tan

1dx

x x

+

(a) 2ln (b) 2ln (c) 22

ln

+ (d) 22

ln

.

96. The value of , 0b

a

xdx a b

x<


21/36

99. The value of3

2

21 x dx

is

(a)1

3(b)

14

3(c)

7

3(d)

28

3.

100. If ( ) ( )

2

01 ,f x x then f x dx= is(a) 1 (b) 0 (c) 2 (d) -2.

101. If ( ) ( )/ 2

0 0sin sin ,xf x dx A f x dx then A

= is

(a) 2 (b) (c)4

(d) 0.

102. ( ) ( )/ 2

0sin cos log sin cosx x x x dx

+ =

(a) -1 (b) 1 (c) 0 (d) None of these.

103. The function ( )1

x dtL x

t= satisfies the equation

(a) ( ) ( ) ( )L x y L x L y+ = + (b) ( ) ( )x

L L x L yy

= +

(c) ( ) ( ) ( )L xy L x L y= + (d) None of these.

104. The value of integral21

0

xe dx lies in the interval

(a) ( 0,1 ) (b) ( -1,0 ) (c) ( 1, e ) (d) None of these.

105. If ( ) ( )3

2 2

0 0cos cos ,P f x dx and Q f x dx then

= =

(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

( ) ( )

3 32 2

0 13 2 3 2 ,ax bx c dx ax bx c dx then+ + = + + (a) a + b + c = 3 (b) a + b + c = 1 (c) a + b + c = 0 (d) a + b + c = 2.

107. ( )2

cos sinpx qx dx

is equal to ( where p and q are integers )

(a) (b) 0 (c) (d) 2 .

108. If ( ) ( )40

cos ,x

g x t dt then g x = + equals

(a) ( ) ( )g x g + (b) ( ) ( )g x g (c) ( ) ( )g x g (d) ( ) ( )/g x g .


22/36


01 xe dx

+ =

(a) -1 (b) 2 (c) 11 e+ (d) None of these.

110.( )

( ) ( )

2

0 2

a f xdx

f x f a x=

+

(a) a (b)2

a(c) 2a (d) 0.

111. The value of0

sinn

x dx +

is

(a) 2 1 cosn + + (b) 2 1 cosn + (c) 2 1n + (d) 2 cosn + .

112. If/ 4

20

tan ,n

n n nu x dx then u u

= + =

(a)1

1n (b)

1

1n +(c)

1

2 1n (d)

1

2 1n +.

113.1

0logsin

2x dx

=

(a) log 2 (b) log2 (c) log22

(d) log2

2

.

114.1

20

log

1

xdx

x=

(a) log22

(b) log2 (c) log2

2

(d) log 2 .

115./ 2

0cotx x dx

equals

(a) log22

(b) log2

2

(c) log2 (d) log2 .

116. The integral value ( ) ( )( )0

3 2

23 3 3 1 cos 1x x x x x dx

+ + + + + + is

(a) 2 (b) 4 (c) 0 (d) 8 .

117. If ( )1

2

sin

11 sin , 0,

2 3xt f t dt x x then f

= equal to

(a) 3 (b)1

3(c)

1

3(d) 3

118.2

0

1sin sin

2

n

x x dx

equals

(a) n (b) 2n (c) - 2n (d) None of these


23/36

119. The value of 31a

adx

x x + is

(a) 0 (b) 60

1

1

a

dxx+ (c) 30

12

1

a

dxx+ (d) ( )30

1

1

a

dxa x+ .

120./ 3

/ 6 1 tandx

x

=+

(a) /12 (b) / 2 (c) / 6 (d) / 4

121.

4

4 4

sin

sin cos

xdx

x x

=

+

(a) / 4 (b) / 2 (c) 3 / 2 (d)

122. If f is continuous function, then

(a) ( ) ( ) ( )2 2

2 0f x dx f x f x dx

= (b) ( ) ( )

5 10

3 62 1f x dx f x dx

=

(c) ( ) ( )5 4

3 41f x dx f x dx

= (d) ( ) ( )

5 6

3 21f x dx f x dx

=

123. The value of2 2 2

1lim .....

1 4 9 2n

n n n

n n n n

+ + + + + + + is equal to

(a)2

(b)

4


124. 3 3 3 31 4 1

lim .....1 2 2n n n n

+ + ++ +

is equal to

(a)1

log 33

e (b)1

log 23

e (c)1 1

log3 3

e (d) None of these.

125.

99 99 99 99

100

1 2 3 ....limn

n

n

+ + +=

(a)9

100

(b)1

100

(c)1

99

(d)1

101

.

126.

1/!

lim

n

nn

n

n

equals

(a) e (b) 1/e (c) / 4 (d) 4 / .

127.

2

2 21

1lim

n

nr

requals

n n r = +

(a) 1 5+ (b) 1 5 + (c) 1 2 + (d) 1 2+ .


24/36

128.1 1 1 1

lim .....1 2 2n n n n n

+ + + = + +

(a) 0 (b) log 4e (c) log 3e (d) log 2e .

129. 2 21

lim

n

nk

k

n k = + is equal to

(a)1

log22

(b) log2 (c) / 4 (d) / 2 .

130.( )2 2 2

1 1 1 1lim .....

2 1n n n n n n n n n

+ + + + + + +

is equal to

(a) 2 2 2+ (b) 2 2 2 (c) 2 2 (d) 2.

131. 1

1 2 3 ......lim

p p p p

pn

n

n +

+ + + +=

(a)1

1p + (b)1

1 p (c)1 1

1p p

(d)1

2p + .

132.1

1lim

rn

n

nr

en =

is equal to

(a) 1e + (b) 1e (c) 1- e (d) e.

133. The correct evaluation of4

0

sin x dx

is

(a)8

3

(b)

2

3

(c)

4

3

(d)

3

8

.

134. The points of intersection of ( ) ( )12

2 5x

F x t dt = and ( )2 0 2 ,x

F x tdt = are

(a)6 36

,5 25

(b)2 4

,3 9

(c)1 1

,3 9

(d)1 1

,5 25

.

135. ( )0

"b c

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 ( )1

x

F x t dt = = on the interval1 1

,2 2

is given by

(a)3

8(b)

1

2 (c)

3

8 (d)

2

5.

137. ( )/ 2

2 2

/ 2sin cos sin cosx x x x dx

+ =

(a)2

15(b)

4

15(c)

6

15(d)

8

15.


25/36

138.

( )30

2 1

dx

x x

=

+ +

(a)3

8(b)

1

8(c)

3

8 (d) None of these.

139. If ( )2

2

2

1

,x

t

x

f x e dt+ =

then ( )f x increases in

(a) ( )2, 2 (b) No value ofx (c) ( )0, (d) ( ), 0

40. If ( ) ( )log

,x

f x dx xe f x= + then ( )f x is

(a) 1 (b) 0 (c) xce (d) log x .

141. The value of ( )3/ 2

0sin cos d

is

(a) 2/9 (b) 2/15 (c) 8/45 (d) 5/2.

142. 201

log1

dxx

x x

+ + is equal to

(a) log2 (b) log2 (c) ( )/ 2 log 2 (d) ( )/ 2 log 2 .

143.

( )20 2

1

x ln x dx

x

+ is equal to

(a) 0 (b) 1 (c) (d) None of these.

144. If ( ) 2 ,1

t

t

dxf t

x=

+ then ( )' 1f is

(a) Zero (b) 2 / 3 (c) -1 (d) 1.

145. If ( ) ( )3

2log , 0 ,

x

xF x t dt x= >

then ( )F x =

(a) ( )29 4 logx x x (b) ( )24 9 logx x x (c) ( )29 4 logx x x+ (d) None of these.

146.1

1

20

2sin

1

d xdx

dx x

+ is equal to

(a) 0 (b) (c) / 2 (d) / 4 .

147. Let ( ) 21

2 .x

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

(a) 1 (b)1

2 (c) 1

2 (d) 0 and 1.


26/36

148. ( )( )20 1 1xdx

x x

=

+ +

(a) 0 (b) / 2 (c) / 4 (d) 1.

149. Let ( ) ( ) ( )3

sin4

sin

13; 0. 1 ,

x

xd eF x x If e dx F k F dx x x = > = then one of the possible value of k, is

(a) 15 (b) 16 (c) 63 (d) 64.

150. If ( ) ( )0

sin , 'x

f x t t dt then f x= =

(a) cos sinx x x+ (b) sinx x (c) cosx x (d) None of these.

151.2 2 2

2 2 2 2

1 1 2 4 1lim sec sec .... sec 1n n n n n n

+ + + equals

(a) tan1 (b)1

tan12

(c)1

sec12

(d)1

cos 12

ec .

152. Area bounded by the curve log ,y x= x axis and the ordinates 1, 2x x= = 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 24 ,y x y axis= and the lines 1, 4y y= = is

(a) 3 .sq unit (b)7

.5

sq unit (c)7

.3

sq unit (d) None of these.

154. If the ordinate x a= divides the area bounded by the curve2

81 ,y x axis

x

= +

and the ordinates

2, 4x x= = 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 0x=

and2

x=

, is

(a) 0 (b) 2 .sq unit (c) .sq unit (d) 4 .sq unit .

156. Area under the curve 2 cos 2y s in x x= + between 0x = and ,4

x

= is

(a) 2 .sq unit (b) 1 .sq unit (c) 3 .sq unit (d) 4 .sq unit.

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

(a) 56 .9

sq unit (b) 64 .9

sq unit (c) 8 .sq unit (d) None of these.


27/36

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

(a)4

3(b) 1 (c)

2

3(d)

1

3.

159. Area bounded by lines 2 , 2y x y x= + = and 2x = is

(a) 3 (b) 4 (c) 8 (d)16.

160. The ra tio of the areas bounded by the curves cosy x= and cos2y x= between 0, / 3x x = =

and ,x axis is

(a) 2 :1 (b) 1: 1 (c) 1 : 2 (d) 2 : 1.

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

(a)15

.2

sq unit (b)15

.4

sq unit (c)17

.2

sq unit (d)17

.4

sq unit.

162. The area bounded by the x axis and the curve siny x= and 0x = , x = is

(a) 1 (b) 2 (c) 3 (d) 4.

163. For 0 ,x the area bounded by y x= and sin ,y x x= + is

(a) 2 (b) 4 (c) 2 (d) 4 .

164. The area of the region bounded by the x axis and the curves defined by ( )tan , / 3 / 3y x x= is

(a) log 2 (b) log 2

(c) 2log2 (d) 0.

165. If the area above the x axis , bounded by the curves 2kxy = and 0x = and 2x = is3

,2In

then the value of k is

(a)1

2(b) 1 (c) -1 (d) 2.

166. The area bounded by the x-axis, the curve ( )y f x= and the lines 1,x x b= = is equal to 2 1 2b + for

all 1,b > then ( )f x is

(a) 1x (b) 1x + (c) 2 1x + (d) 21

x

x+.


28/36

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

then ( )f x is

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

(c) ( ) ( ) ( )1 cos 3 4 3sin 3 4b x x + + + (d) None of these.

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

(a) 1 (b)2

3(c)

4

3(d)

8

3.

169. Area under the curve 2

4y x x= within the

x axisand the line 2x

=, is

(a)16

.3

sq unit (b)16

.3

sq unit (c)4

.7

sq unit (d) Cannot be calculated.

170. Area bounded by the curve 3 2 10 0,xy x y x axis = and the lines 3, 4x x= = is

(a) 16 log 2 13 (b) 1 6 lo g 2 3 (c) 16 log 2 3+ (d) None of these.

171. The area bounded by curve 2 , 4y x line y= = 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 0, 2x x= = and the curves 22 , 2xy y x x= = is

(a)4 1

3 log 2 (b)

3 4

log 2 3+ (c)

41

log2 (d)

3 4

log 2 3 .

173. The area between the curve 2sin ,y x x axis= and the ordinates 0x = and2

x

= is

(a)2

(b)

4

(c)

8

(d) .

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

(a)2

(b)

4

(c)

3

(d) .

175. The area bounded by the curve 24y x x= and the x axis , is

(a)30

.7

sq unit (b)31

.7

sq unit (c)32

.3

sq unit (d)34

.3

sq unit.


29/36

176. Area of the region bounded by the curve tan ,y x= tangent drawn to the curve at4

x

= and the x axis is

(a)1

4(b)

1log 2

4+ (c)

1log 2


177. The area between the curve 24 3y x x= + and x axis is

(a) 125 / 6 (b) 125 / 3 (c) 125 / 2 (d) None of these.

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

(a) 24a (b) 28a (c)228

3a (d)

235

3a .

179. The area of the region bounded by 1y x= and 1y = is

(a) 2 (b) 1 (c)1

2(d) None of these.

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

(a)24

3a (b)

28

3a (c)

22

3a (d)

25

3a .

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

(a) 2a (b) 22 a (c) 23 a (d) 24 a .

182.The area enclosed by the parabolas 2 1y x= and 21y x= 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 2 2 9x y+ = by 1x = is

(a) ( )11

9 sec 3 82

(b) ( )1

9sec 3 8

(c) ( )18 9sec 3 (d) None of these.

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

(a) 4 (b) 2 (c) 3 (d) 1.

185. The area bounded by the curves loge

y x= and ( )2

logey x= is

(a) 3 e (b) 3e (c) ( )1

32

e (d) ( )1

32

e .


30/36

186. The area of figure bounded by ,x xy e y e= = and the straight line 1x = is

(a)1

ee

+ (b)1

ee

(c)1

2ee

+ (d)1

2ee

+ + .

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

(a) 9 (b)2 7

4(c) 36 (d) 18 .

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

(a) 3 (b) 4 (c) 1 (d) 2 .

189. The parabolas 2 4y x= and 2 4x y= divide the square region bounded by the lines 4x = , 4y = and the

coordinate axes. If 1 2 3, ,S S S are respectively the areas of these of these parts numbered from top to bottom

, then 1 2 3: :S S S 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 3 4,y x x axis= + and the line 1x = and B is that area

bounded by curve

2

3 4,y x x axis= + = and the lines 1x = and 4x = 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 ( ) ( )2 2

1 , 1y x y x= + = and then line 14

y = is

(a) 1/6 (b) 2/3 (c) 1/4 (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 ,4 4

x x

= = > is sin cos 24

+ +

then

2f

is

(a) 1 24

(b) 1 24

+

(c) 2 14

+

(d) 2 14

+

.

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.


31/36

LEVEL - 3 (Tougher Problems)

1. Let ( )f x be a function satisfying ( )f x = ( )f x with ( )0 1f = and ( )g x be the function satisfying

( ) ( ) 2f x g x x+ = . The value of integral ( ) ( )1

0

f x g x dx is equal to

(a) ( )1

74

e (b) ( )1

24

e (c) ( )1

32

e (d) None of these.

2. Let f be a positive function . Let1 2

1 1( (1 )) , ( (1 ))

k k

k kI xf x x dx I f x x dx

= =

when 2 1 0.k > Then 1 2/I I is

(a) 2 (b) k (c)1/2 (d) 1.

3.

71

401

xdx

x is equal to

(a) 1 (b)1

3(c)

2

3(d)

3

.

4. If n is any integer, then ( )2cos 3

0cos 2 1xe n xdx

+ =

(a) x (b) 1 (c) 0 (d) None of these.

5. Let a, b, c be non-zero real numbers such that ( )( ) ( )( )1 2

8 2 8 2

0 01 cos 1 cosx ax bx c dx x ax bx c dx+ + + = + + + .

Then the quadratic equation 2 0ax bx c+ + = 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 ( )1

, 1,x

f x t dt x

= then

(a) f and 'f are continous for 1 0x + > (b) f is continous but f is not continous for 1 0x + >

(c) f and f are not continous at 0x = (d) f is continous at 0x = but f is not so.

7. Let ( ) ( )0

x

g x f t dt = where ( ) [ ]1 1, 0,12

f t t and ( )1

02

f t for ( ]1,2 ,t then

(a) ( )3 1

22 2

g < (b) ( )0 2 2g < (c) ( )3 5

22 2

g< (d) ( )2 2 4g< < .

8. The value of2cos

, 0,

1x

xdx a

a

>

+is

(a) (b) a (c) 2

(d) 2 .


32/36

9. If ( ) ( )( )( )

( )

1, 1 ,1

xf a

x f a

ef x I xg x x dx

e = =

+ and ( )( )( )( )

2 1 ,f a

f aI g x x dx

= then the value of 2

1

I

Iis

(a) 1 (b) 3 (c) 1 (d) 2.

10. Let ( ) ( )1 1

0 01, ,f x dx x f x dx a= = and ( )

12 2

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

1 2

0x a f x dx =

(a) 0 (b) 2a (c)2 1a (d) 2 2 2a a + .

11. Given that( )( )( ) ( )( )( )

2

2 2 2 2 2 20 2

x dx

a b b c c ax a x b x c

=

+ + ++ + + then the value of ( )( )2

2 20 4 9

x dx

x x

+ + is

(a)60

(b)

20

(c)

40

(d)

80

.

12. If ( ) ( )1

0, 1 ,

nml m n t t dt = + then the expression for ( ),l m n in terms of ( )1, 1l m n+ is

(a) ( )2 1, 11 1

n nl m n

m m +

+ +(b) ( )1, 1

1

n l m nm

+ +

(c) ( )2

1, 11 1

n nl m n

m m+ +

+ +(d) ( )1, 1

1

ml m n

n+

+

13.4 4 4 3 3 3

5 5

1 2 3 ..... 1 2 3 ....lim limn n

n n

n n

+ + + + + + + + =

(a)1

30(b) Zero (c)

1

4(d)

1

5.

14. If ( )2

5

0

2 , 0,5

txf x dx t t= > then 425f =

(a)2

5(b)

5

2(c)

2


15. For which of the following values of m , the area of the region bounded by the curve2

y x x= and the

line y mx= equals9

2

(a) - 4 (b) - 2 (c) 2 (d) 4.

16. Area enclosed between the curve ( )2 32y a x x = and line 2x a= above x axis is

(a) 2a (b)2

3

2

a(c) 22 a (d) 23 a .

17. What is the area bounded by the curves 2 2 9x y+ = and 2 8y x= is

(a) 0 (b)12 2 9 1

9sin3 2 3

+


18. The area bounded by the curves 1y x= and 1y x= + is

(a) 1 (b) 2 (c) 2 2 (d) 4.


33/36

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

[ ]3 / 2

/ 2

2sin x dx

is

(a) (b) 0 (c)2

(d)

2

.

20. If ( ) 1sin , ' 22 2xf x A B f = + =

and ( )

1

0

2 ,Af x dx

= then the constants A and B are respectively

(a)2 2

and

(b)2 3

and

(c)4

0and

(d)4

0 and

.

21./ 4

0tan ,

n

nI x dx

= then [ ]2lim n nn I I + equals

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

22. The area bounded by the curves , ,y In x y In x y in x= = = and y in x= is

(a) 4 .sq unit (b) 6 .sq unit (c) 10 .sq unit (d) None of these.

23. ( )0

1sin

2,

sin

n x

dx n N x

+ equals

(a) n (b) ( )2 12

n

+ (c) (d) 0.

24. If ( )21

00,xe x dx = then

(a) 1 2< < (b) 0 < (c) 0 1< < (d) None of these.

25.10

sin x dx

is

(a) 20 (b) 8 (c) 10 (d) 18.

26.( )

2

2 1 sin

1 cos

x xdx

x

+

+is

(a) 2 / 4 (b) 2 (c) 0 (d) / 2 .

27. If2 2 31 1 2 23

1 2 3 40 0 1 1

2 , 2 , 2 , 2 ,x x xI dx I x dx I dx I dx= = = = then

(a) 3 4I I= (b) 3 4I I> (c) 2 1I I> (d) 1 2I I> .

28. If ( )1

2 3 ,f x f xx

=

then ( )2

1f x dx is equal to

(a)3

25

In (b) ( )3

1 25

In

+ (c)3

25

In

(d) None of these.


34/36

LEVEL - 1 (Fundamentals of Definite Integration)

ANSWER KEY

1. c

2. a

3. c

4. b

5. c

6. a

7. b

8. d

9. a

10. c

11. d

12. c

13. a

14. d

15. a

16. b

17. c

18. d

19. d

20. c

21. b

22. a

23. c

24. b

25. a

26. c

27. c

28. b

29. c

30. d

31. d

32. a

33. b

34. b

35. d

36. b

37. a

38. a

39. c

40. c

41. b

42. b

43. a

44. b

45. a

46. c

47. b

48. c

49. b

50. a

51. d

52. a

53. a

54. a

55. b

56. c

57. b

58. c

59. b

60. d

61. b

62. a

63. d

64. c

65. a

66. c

67. a

68. d

69. d

70. b

71. b

72. a

73. d

74. c

75. a

76. c

77. b

78. a

79. d

80. b

81. b

82. c

83. d

84. a

85. a

86. b

87. b

88. a

89. a

90. b

91. c

92. c


35/36

LEVEL - 2 (Properties of Definite Integration)

ANSWER KEY

1. b

2. c

3. d

4. b

5. b

6. d

7. a

8. c

9. d

10. c

11. d

12. c

13. d

14. a

15. a

16. d

17. b

18. c

19. b

20. b

21. c

22. d

23. b

24. d

25. b

26. c

27. b28. b

29. d

30. a

31. a

32. b

33. b

34. a

35. c

36. c

37. d

38. a

39. b

40. a

41. b

42. a

43. d

44. d

45. c

46. b

47. c

48. d

49. d

50. c

51. a

52. b

53. b

54. a

55. c56. b

57. c

58. a

59. b

60. c

61. a

62. b

63. a

64. b

65. c

66. b

67. a

68. b

69. d

70. d

71. c

72. a

73. b

74. c

75. a

76. c

77. a

78. a

79. d

80. b

81. c

82. b

83. a84. c

85. c

86. c

87. c

88. a

89. b

90. c

91. c

92. a

93. d

94. d

95. d

96. b

97. c

98. c

99. d

100. a

101. b

102. c

103. c

104. c

105. c

106. c

107. d

108. a

109. d

110. a

111. b112. a

113. a

114. c

115. b

116. b

117. a

118. b

119. a

120. a

121. d

122. d

123. b

124. b

125. b

126. b

127. b

128. d

129. a

130. b

131. a

132. b

133. d

134. a

135. b

136. c

137. b

138. a

139. d140. c

141. c

142. a

143. a

144. d

145. a

146. c

147. a

148. c

149. d

150. b

151. b

152. c

153. c

154. b

155. d

156. b

157. d

158. a

159. b

160. d

161. b

162. b

163. a

164. c

165. b

166. d

167. a168. b

169. a

170. c

171. b

172. d

173. b

174. c

175. c

176. d

177. a

178. c

179. b

180. b

181. a

182. d

183. b

184. d

185. a

186. c

187. a

188. c

189. b

190. a

191. d

192. b

193. c


36/36

LEVEL - 3 (Tougher Problems)

ANSWER KEY

1. d

2. c

3. b

4. c

5. b

6. a

7. b

8. c

9. d

10. a

11. a

12. a

13. d

14. a

22. a

23. c

24. c

25. d

26. b

27. d

28. b

15. b

16. b

17. b

18. b

19. c

20. c

21. b

definite integration assignment for iit-jee

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