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9 - INTEGRAL CALCULUS

Page 1

( Answers at the end of all questions ) 1

(1)

1

2

∫

If I 1 =

2 x dx ,

I2 =

0

∫

2

2 x dx ,

2

I4 =

1

(b) I1 > I2

(c) I3 = I4

∫

3

2 x dx , then

1

(d) I3 > I4

[ AIEEE 2005 ]

The area enclosed between the curve y = log e ( x + e ) and the coordinate axes is (a) 1

(3)

2

I3 =

0

(a) I2 > I1

(2)

∫

3

2 x dx ,

(b) 2

(c) 3

(d) 4

2

[ AIEEE 2005 ]

2

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

(b) 1:3:1

(c) 2:1:2

(d) 1:1:1

[ AIEEE 2005 ]

2

(4)

∫

 ( log x - 1 )    dx  1 + ( log x ) 2 

(a) (c)

(5)

log x ( log x ) 2 + 1

xe x 1+ x

+ c

(b)

+ c

2

(d)

x x2 + 1

+ c

x ( log x ) 2 + 1

+ c

[ AIEEE 2005 ]

Let f ( x ) be a non-negative continuous function such that the area bounded by the π π curve y = f ( x ), X-axis and the ordinates x = and x = β > is 4 4 π π ( β sin β + cos β + 2 β ). Then f ( ) is 4 2 (a)

π + 4

(c) 1 -

2 -1 π 4

The value of

∫

-π

(a) aπ

(b)

(b)

π 4

cos 2 x 1 + ax

π 2

dx ,

(c)

2 + 1 π + 4

(d) 1 -

2 π

(6)

is equal to

2

[ AIEEE 2005 ]

a > 0 is

π a

(d) 2π

[ AIEEE 2005 ]

9 - INTEGRAL CALCULUS

Page 2

( Answers at the end of all questions ) n

(7)

∑

lim

n→∞ r =1

If

is

(b) e - 1

(a) e

(8)

r n e

sin x dx - α)

∫ sin ( x

(c) 1 - e

∫ cos x

dx - sin x

( b ) ( cos α, sin α ) ( d ) ( - cos α, sin α )

π  x -  + C log tan  2 8   2

 x 3π  log tan   + C 8   2 2

3

( 10 ) The value of

∫

1 - x 2 dx

 x  log cot   + C  2  2

1

(b)

1

(c)

[ AIEEE 2004 ]

is equal to

1

(a)

[ AIEEE 2004 ]

= Ax + B log sin ( x - α ) + C, then the value of ( A, B ) is

( a ) ( sin α, cos α ) ( c ) ( - sin α, cos α )

(9)

(d) e + 1

3π   x + log tan   + C 8   2 2

1

(d)

[ AIEEE 2004 ]

is

-2

28 3

(a)

(b)

14 3

(c)

7 3

(d)

1 3

[ AIEEE 2004 ]

π 2

( 11 ) The value of I =

∫

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

0

(a) 0

(b) 1

(c) 2

∫

is

(d) 3

[ AIEEE 2004 ]

π

π ( 12 ) If

dx

2

x f ( sin x ) dx = A

0

(a) 0

∫

f ( sin x ) dx , then A is equal to

0

(b) π

(c)

π 4

(d) 2π

[ AIEEE 2004 ]

9 - INTEGRAL CALCULUS

Page 3

( Answers at the end of all questions ) f(a)

ex

( 13 ) If f ( x ) =

1 + ex

,

∫

I1 =

f(a)

x g { x ( 1 - x ) } dx

I2 =

and

f(- a)

∫

g { x ( 1 - x ) } dx ,

f(- a)

I then the value of 2 is I1 (b) -3

(a) 2

(c) -1

(d) 1

[ AIEEE 2004 ]

( 14 ) The area of the region bounded by the curves y = l x - 2 l, X-axis is (a) 1

(b) 2

(c) 3 x2

∫

( 15 ) The value of (a) 3

lim

x→0

(b) 2

x = 1,

x = 3

and

(d) 4

[ AIEEE 2004 ]

(d) 0

[ AIEEE 2003 ]

sec 2 t dt

0

is

x sin x (c) 1 1

( 16 ) The value of the integral I =

∫

x ( 1 - x )n dx

is

0

(a)

1 n + 1

(b)

1 n + 2

(c)

1 1 n + 1 n + 2 t

y

( 17 ) If f ( y ) = e ,

g ( y ) = y,

y > 0

and

F(t) =

1 1 + n + 1 n + 2 [ AIEEE 2003 ]

(d)

∫

f ( t - y ) g ( y ) dy ,

then

0

(a) F(t) = te (c) F(t) = e

t

t

(b) F(t) = te-

t t

(d) F(t) = 1 - e- (1 + t)

- (1 + t)

[ AIEEE 2003 ]

b

( 18 ) If f ( a + b - x ) = f ( x ),

then the value of

∫

x f ( x ) dx

is

a

(a)

(c)

b -a 2

b

∫

f ( x ) dx

(b)

a

a + b 2

b

∫

a

f ( b - x ) dx

a + b 2 b

(d)

∫

a

b

∫

f ( x ) dx

a

f ( a + b - x ) dx

[ AIEEE 2003 ]

9 - INTEGRAL CALCULUS

Page 4

( Answers at the end of all questions ) ( 19 ) Let

4

d e sin x F(x ) = , dx x

x > 0.

3 sin x3 e dx = F ( K ) - F ( 1 ), then one of the x

∫

If

1

possible values of K is ( a ) 15

( b ) 16

( c ) 63

( d ) 64

[ AIEEE 2003 ]

( 20 ) Let f ( x ) be a function satisfying f ’ ( x ) = f ( x ) with f ( 0 ) = 1 and g ( x ) be a 1

2

function that satisfies

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

∫

f ( x ) g ( x ) dx

0

is (a) e -

e2 5 2 2

(b) e +

e2 2

-

3 2

(c) e -

e2 3 2 2

(d) e +

e2 2

+

5 2

( 21 ) If

∫ x sin x dx

= - x cos x + α, then the value of α is

( a ) sin x + c ( c ) x cos x + c

( b ) cos x + c ( d ) cos x - sin x + c 1 - cos 2x dx + 1

∫ cos 2x

( 22 ) The value of

[ AIEEE 2003 ]

( a ) tan x - x + c ( c ) x - tan x + c

[ AIEEE 2002 ]

is

( b ) x + tan x + c ( d ) - x - cot x + c

[ AIEEE 2002 ]

π 2

∫

( 23 ) The value of

a cos x + b 2 sin 2 x 0

2

2

(a ) πab

( b ) π ab

∫e

( 24 ) The value of

( a ) log ( x

dx 2

4

( c ) 3 log ( x

3 log x

( c ) π / ab

( x 4 + 1 ) -1 dx

+ 1) +c

(b)

4

(d)

+ 1) +c

is

(d)

π / 2ab

[ AIEEE 2002 ]

is

1 4 log ( x + 1 ) + c 4 - log ( x4 + 1 ) + c

[ AIEEE 2002 ]

9 - INTEGRAL CALCULUS

Page 5

( Answers at the end of all questions ) log x

∫

( 25 ) The value of

x2

dx is

( a ) log ( x + 1 ) + c

(b)

( c ) log ( x - 1 ) + c

(d)

sin2 x

∫

( 26 ) The value of

sin

-1

-

1 log ( x + 1 ) + c x 1 log ( x + 1 ) + c 2 cos2 x

( t ) dt +

0

π 2

(a)

[ AIEEE 2002 ]

∫

cos - 1 ( t ) dt

is

0

(b) 1

(c)

π 4

(d)

π

[ AIEEE 2002 ]

( 27 ) If the area bounded by the X-axis, the curve y = f ( x ) and the lines x = 1, x = b is b2 + 1 -

equal to x - 1

(a)

∫ [x 0

( 28 )

2

for all b > 1, then f ( x ) is

x + 1

(b)

(c)

x2 + 1

(d)

x 1 + x2

[ AIEEE 2002 ]

]

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

-2

(a) 4

(c) -1

(b) 0

(d) 1

[ IIT 2005 ]

2

2

( 29 ) Find the area between the curves y = ( x - 1 ) , y = ( x + 1 ) and y = 1 3

(a)

1

∫t

( 30 ) If

(b)

2 3

2 f ( t ) dt = 1

(c)

4 3

(d)

1 6

1 4 [ IIT 2005 ]

- sin x, x ∈ [ 0, π / 2 ], then f ( 1 /

3 ) is

sin x

(b) 1/3

(a) 3 t2

( 31 ) If

∫ x f ( x ) dx 0

(a) -

2 5

=

2 5 t 5

(b) 0

(c) 1

(d)

 4  for t > 0, then f    25 

(c)

2 5

(d) 1

[ IIT 2005 ]

3

is

[ IIT 2004 ]

9 - INTEGRAL CALCULUS

Page 6

( Answers at the end of all questions ) 1

( 32 )

∫

0

(a)

1-x dx is equal to 1+ x

π + 1 2

(b)

π - 1 2

(c) 1

(d)

( 33 ) If the area bounded by the curves x = ay 1

(a)

1 3

(b)

3

x2 + 1

∫

( 34 ) If f ( x ) =

1 2

(c)

2

π

[ IIT 2004 ]

2

and y = ax is 1, then a is equal to

(d) 3

[ IIT 2004 ]

2

e - t dt , then the interval in which f ( x ) is increasing is

x2

( a ) ( 0, ∞ )

(b) (- ∞, 0) 1

( 35 ) If I ( m, n ) =

∫t

( c ) [ - 2, 2 ]

m ( 1 + t )n dt ,

( d ) nowhere

[ IIT 2003 ]

m, n ∈ R, then I ( m, n ) is

0

(a)

n I [ ( m + 1) , ( n - 1 ) ] 1+ m

(b)

2n m I [ ( m + 1) , ( n - 1 ) ] 1+ m 1+ n

(c)

2n m I [ ( m + 1) , ( n - 1 ) ] 1+ m 1+ m

(d)

m I [ ( m + 1) , ( n - 1 ) ] n+1

( 36 ) Area bounded by the curves y = (a) 2

( b ) 18

3

[ IIT 2003 ]

x , x = 2y + 3 in the first quadrant and X-axis is

(c) 9

34 3

(d)

[ IIT 2003 ]

( 37 ) The area bounded by the curves y = l x l - 1 and y = - l x l + 1 is (a) 1

(b) 2 x

( 38 ) If f ( x ) =

∫

(c) 2

2

(d) 4

2 2 - t 2 dt , then the real roots of the equation x

[ IIT 2002 ]

- f ’ ( x ) = 0 are

1

(a) ± 1

(b)

±

1 2

(c)

±

1 2

( d ) 0 and 1

[ IIT 2002 ]

9 - INTEGRAL CALCULUS

Page 7

( Answers at the end of all questions ) ( 39 ) Let T > 0 be a fixed real number. Suppose f is a continuous function such that for 3 + 3T

T

all x ∈ R, f ( x + T ) = f ( x ). If I =

∫

f ( x ) dx , then the value of

0

(a)

3 I 2

(b)

( c ) 3I

I

1 2

-

1 2

(a) -

  [ x 1 

∫

( 40 ) The integral equals

]+

∫ f ( 2x ) dx

is

3

( d ) 6I

[ IIT 2002 ]

 1+ x  ln    dx equals  1- x 

2

(b) 0

(c) 1

( d ) 2 ln

1 2

[ IIT 2002 ]

x

∫ f ( t ) dt

( 41 ) If f : ( 0, ∞ ) → R, F ( x ) =

2

and

2

F ( x ) = x ( 1 + x ),

then f ( 4 ) equals

0

(a)

5 4

(b) 7

π ( 42 ) The value of

(a) π

( 43 ) If f ( x ) =

(a ) 0

∫

(c) 4

cos 2 x

1 + ax -π

(b) aπ

dx,

(c)

(d) 2

a > 0, is

π 2

(c) 2 e2

( 44 ) The value of the integral

∫

e- 1

(a)

3 2

( d ) 2π

 e cos x sin x for l x l ≤ 2,   2 otherwise,

(b) 1

(b)

5 2

[ IIT 2001 ]

(c) 3

[ IIT 2001 ]

3

then

(d) 5

=

-2

(d) 3

log e x dx x

∫ f ( x ) dx

[ IIT 2000 ]

is

[ IIT 2000 ]

9 - INTEGRAL CALCULUS

Page 8

( Answers at the end of all questions ) ( 45 ) If f ( x ) = ∫ e x ( x - 1 ) ( x - 2 ) dx , then f decreases in the interval ( a ) ( - ∞, - 2 )

( 46 )

Let

g(x) =

0 ≤ f(t) ≤

(a)

-

(c)

( 47 )

1 2

( b ) ( - 2, - 1 ) x ∫ f ( t ) dt , 0

( c ) ( 1, 2 )

where f is such that

( d ) ( 2, + ∞ )

1 ≤ f(t) ≤ 1 2

for

[ IIT 2000 ]

t ∈ [ 0, 1 ]

and

for t ∈ [ 1, 2 ]. Then g ( 2 ) satisfies the inequality

3 1 ≤ g(2) ≤ 2 2 3 5 ≤ g(2) ≤ 2 2

( b ) 0 ≤ g(2) ≤ 2 (d) 2 < g(2) < 4

[ IIT 2000 ]

If for a real number y, [ y ] is the greatest integer less than or equal to y, then the 3π 2

value of the integral

∫

[ 2 sin x ] dx is

π 2

(a) -π

(b) 0

(c) -

π 2

(d)

π 2

[ IIT 1999 ]

(d) -

1 2

[ IIT 1999 ]

3π

4 ( 48 )

∫

π 4

dx 1 + cos x

(a) 2

( 49 )

=

(b) -2

(c)

1 2

For which of the following values of

m, is the area of the region bounded by the 9 ? curve y = x - x and the line y = mx equals 2 2

(a) -4

( 50 )

(b) -2

(c) 2

(d) 4

[ IIT 1999 ]

If f ( x ) = x - [ x ], for every real number x, where [ x ] is the integral part of x, then 1

∫ f ( x ) dx

is

(a) 1

(b) 2

-1

(c) 0

(d)

1 2

[ IIT 1998 ]

9 - INTEGRAL CALCULUS

Page 9

( Answers at the end of all questions ) x

∫ cos

( 51 ) If g ( x ) =

4

t dt , then g ( x +

π ) equals

0

(a) g(x) + g(π)

(b) g(x) - g(π)

(c) g(x)g(π)

(d)

g( x ) g( π)

k

( 52 )

∫

Let f be a positive function. If I1 =

k

x f [ x ( 1 - x ) ] dx

and I2 =

1- k

where

I1 I2

2k - 1 > 0, then

(a) 2

(b) k

(c)

[ IIT 1997 ]

∫ f [ x ( 1 - x ) ] dx ,

1- k

is

1 2

(d) 1

[ IIT 1997 ]

( 53 ) The slope of the tangent to a curve y = f ( x ) at [ x, f ( x ) ] is 2x + 1. If the curve passes through the point ( 1, 2 ), then the area of the region bounded by the curve, the X-axis and the line x = 1 is (a)

5 6

6 5

(b)

1 6

(c)

(d) 6

[ IIT 1995 ]

2π ( 54 ) The value of

∫

[ 2 sin x ] dx

where [ . ] represents the greatest integer function, is

π (a)

-

5π 3

(b) -π

π 2 ( 55 ) The value of

(a) 0

dx

∫

1 + tan 3 x 0

(b) 1

(c)

(c)

5π 3

(d) -2π

[ IIT 1995 ]

is

π 2

(d)

π 4

[ IIT 1993 ]

( 56 ) If f : R → R be a differentiable function and f ( 1 ) = 4, then the value of f(x) 2t lim dt is ∫ x - 1 x →1 4 (a) 8f’(1)

(b) 4f’(1)

(c) 2f’(1)

(d) f’(1)

[ IIT 1990 ]

9 - INTEGRAL CALCULUS

Page 10

( Answers at the end of all questions ) ( 57 )

If f : R → R and g : R → R are continuous functions, then the value of the integral π 2

∫ -

[ f ( x ) + f ( - x ) ] [ g ( x ) + g ( - x ) ] dx

is

π 2

(a) π

(b) 1

(c) -1

(d) 0 π

∫

( 58 ) For any integer n, the integral

[ IIT 1990 ]

2 e cos x cos 3 ( 2n + 1 ) x dx

has the value

0

(a) π

(b) 1

(c) 0

( d ) none of these

[ IIT 1985 ]

π 2

∫

( 59 ) The value of the integral

0

(a)

( 60 )

π 4

(b)

π 2

cot x cot x +

(c) π

tan x

dx

is

( d ) none of these

[ IIT 1983 ]

If the area bounded by the curves y = f ( x ), the X-axis and the ordinates x = 1 and x = b is ( b - 1 ) sin ( 3b + 4 ), then f ( x ) is ( a ) ( x - 1 ) cos ( 3x + 4 ) ( c ) sin ( 3x + 4 )

( b ) sin ( 3x + 4 ) + 3 ( x - 1 ) cos ( 3x + 4 ) ( d ) none of these

1

( 61 ) The value of the definite integral

∫

[ IIT 1982 ]

2

( 1 + e - x ) dx is

0

(a) -1

(b) 2

(c) 1 + e–

1

( d ) none of these

[ IIT 1981 ]

9 - INTEGRAL CALCULUS

Page 11

( Answers at the end of all questions )

Answers 1 b

2 a

3 d

4 d

5 d

6 b

7 b

8 b

9 a,d

10 a

11 2

12 b

13 a

14 a

15 c

16 c

17 c

18 b

19 d

20 b

21 a

22 c

23 d

24 b

25 b

26 c

27 d

28 a

29 a

30 a

31 c

32 b

33 a

34 b

35 c

36 b

37 b

38 a

39 c

40 a

41 c

42 c

43 c

44 b

45 c

46 b

47 c

48 a

49 b,d

50 a

51 a

52 c

53 a

54 a

55 d

56 a

57 d

58 c

59 a

60 b

61 d

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80