9 - INTEGRAL CALCULUS
Page 1
( Answers at the end e nd of all questions que stions ) 1
(1)
If
I1
=
2
1
x2
dx ,
I2
=
2
0
(a)
(2)
>
I2
(b)
dx ,
=
I3
2
>
I1
2
dx ,
I4
1
(c)
I2
=
I3
1
(d)
I4
3
2 x dx , then
=
I3
>
[ AIEEE 2005 ]
I4
(b) 2
(c) 3
(d) 4
2
[ AIEEE 2005 ]
2
The parabolas parabolas y = 4x and x = 4y divide the square region region bounded bounded by the lines x = 4, y = 4 and the the coordinate coordinate axes. If S 1, S2, S3 are respectively the area of these parts numbered from top to bottom, then S 1: S2 : S3 is (b) 1:3:1
1
(a)
( log x )
(c)
x
[ AIEEE 2005 ]
c
x2
1 x
(d)
c
x2
1
(b)
c 1
xe x
(d) 1:1:1
is equal to
dx
2
log x ( log x ) 2
(c) 2:1:2
2
( log x - 1 )
(4)
( log x ) 2
c
[ AIEEE 2005 ]
1
Let f ( x ) be a non-negative non-negativ e continuous function func tion such that the area bounded by the curve y = f ( x ), X-axis and the ordinates ordinates x = (
sin
(a)
+
+
4
(c) 1
(6)
x2
The area enclosed between the curve curve y = log e ( x + e ) and the coordinate axes is
(a) 1:2:1
(5)
2
0 I1
(a) 1
(3)
x3
-
4
cos
4 2
-1
-
2
+
ax
2
dx ,
(c)
2
and x =
>
4
is
) is
- 2 + 1
4
(d) 1
1
(b)
). Then f (
(b)
cos 2 x
The value of
(a) a
2
4
-
4
+
2
[ AIEEE AIEEE 2005 2005 ]
a > 0 is
a
(d) 2
[ AIEEE 2005 ]
9 - INTEGRAL CALCULUS
Page 2
( Answers at the end e nd of all questions que stions ) r en
n
(7)
lim n
r 1
(a) e
(8)
is
sin x sin ( x
If
- 1
(b) e
)
dx
(c) 1
dx cos x 1
(a)
2 1
(c)
2
(d) e + 1
= Ax + B log sin ( x
( a ) ( sin , cos ) ( c ) ( - sin , cos )
(9)
- e
-
[ AIEEE 2004 ]
) + C, then the value of ( A, B ) is
( b ) ( cos , sin ) ( d ) ( - cos , sin )
[ AIEEE 2004 ]
is equal to
sin x
x
log tan
2
log tan
8
x
3
2
8
3
( 10 ) The value of
C C
x 2 dx
1
1
(b)
2 1
(d)
2
x
log cot
log tan
C
2 x
3
2
8
C
[ AIEEE 2004 ]
is
2
28 3
(a)
(b)
14 3
2
( 11 ) The value of
I
=
(c)
(a) 0
(b) 1
(d)
cos x ) 2
( sin x 1
0
7 3
sin 2x
(c) 2
dx
1 3
[ AIEEE 2004 ]
is
(d) 3
[ AIEEE 2004 ]
2
( 12 ) If
x f ( sin x ) dx = A 0
(a) 0
f ( sin x ) dx , then A is equal to 0
(b)
(c)
4
(d) 2
[ AIEEE 2004 ]
9 - INTEGRAL CALCULUS
Page 3
( Answers at the end e nd of all questions que stions ) f ( a )
ex
( 13 ) If f ( x ) =
I1
x g{ x(1
and
x ) } dx
3
(c)
(b) 2
-1
x
[ AIEEE 2004 ]
(a) 3
(b) 2
- 2 l,
x = 1,
x = 3
and
(d) 4
[ AIEEE 2004 ]
(d) 0
[ AIEEE 2003 ]
is
x sin x
0
x ) } dx ,
sec 2 t dt
0
lim
g{ x(1
(d) 1
(c) 3
x2
( 15 ) The value of
=
f ( a )
( 14 ) The area of the region regi on bounded by the curves y = l x X-axis is (a) 1
I2
is
I1
(b)
=
f ( a )
I2
then the value of
(a) 2
,
ex
1
f ( a )
(c) 1
1
( 16 ) The value of the integral
I
=
x (1
x ) n dx
is
0
(a)
1 n
(b)
1
1 n
(c)
2
1 n
1
1
-
n
1
(d)
2
n
1
+
1 n 2 [ AIEEE 2003 ]
t
y
( 17 ) If f ( y ) = e ,
g ( y ) = y,
y > 0
and
F(t) =
f ( t
y ) g ( y ) dy ,
then
0
(a) F(t) = te (c) F(t) = e
t
t
(b) F(t) = te
- (1 + t)
(d) F(t) = 1
t
- e
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
a
b
f ( x ) dx b
b
f ( x ) dx a
b
f ( b a
b 2
a
b 2
(b)
a
x ) dx
(d)
f ( a a
b
x ) dx
[ AIEEE AIEEE 2003 ]
9 - INTEGRAL CALCULUS
Page 4
( Answers at the end e nd of all questions que stions )
( 19 ) Let
d dx
4
e sin x
F( x) =
,
x
x > 0.
3 sin x3 e dx = F ( K ) x
If 1
- F ( 1 ), then one of the
possible values of K is ( a ) 15
( 20 )
( b ) 16
Let f ( x )
( c ) 63
( d ) 64
be a function satisfying
[ AIEEE 2003 ]
f’(x) = f(x)
with f ( 0 ) = 1
be a
1
2
function that satisfies
and g ( x )
f ( x ) + g ( x ) = x . The value of the integral
f ( x ) g ( x ) dx 0
is (a) e (c) e
( 21 ) If
-
e2 2 e
2
2
5
-
(b) e +
2 3
-
(d) e +
2
x sin x dx =
( a ) sin x + c ( c ) x cos x + c
- x cos x +
1
cos 2x
cos 2x
( a ) tan x - x + c ( c ) x - tan x + c
( 23 ) The value of 0
( a ) log ( x
4
( c ) 3 log ( x
(b) (d)
a 2 cos 2 x
[ AIEEE 2003 ]
2
is
[ AIEEE 2002 ]
is
x + tan x + c - x - cot x + c
2
b 2 sin 2 x
ab
(c)
e 3 log x ( x 4
+ 1) +c 4
2
5
+
dx
(b)
( 24 ) The value of
e
2
2
, then the value of
dx
1
2
ab
2
3
-
( b ) cos x + c ( d ) cos x - sin x + c
( 22 ) The value of
(a )
e2
+ 1) +c
1 ) 1 dx
(b) (d)
ab
[ AIEEE 2002 ]
is
(d)
ab
[ 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 e nd of all questions que stions ) log x
( 25 ) The value of
x
2
dx is
( a ) log ( x + 1 ) + c
(b)
- 1) + c
(d)
( c ) log ( x
1 log ( x + 1 ) + c x 1 log ( x + 1 ) + c 2
sin2 x
cos2 x
sin 1 ( t ) dt
( 26 ) The value of
cos 1 ( t ) dt
0
(a)
is
0
(b) 1
2
[ AIEEE 2002 ]
(c)
(d)
4
[ 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
equal to (a)
x
1
1
for all b > 1, then f ( x ) is
2
(b)
x
1
x2
(c)
1
x
(d) 1
x
2
[ AIEEE 2002 ]
0
x3
( 28 )
3x 2
3x
3
(x
1 ) cos ( x
1 ) dx =
2
(a) 4
(b) 0
(c)
-1
(d) 1
( 29 ) Find the area between the curves y = ( x 1 3
(a)
(b)
2 3
(c)
4 3
[ IIT 2005 ]
- 1 )2, y = ( x + 1 )2 and y =
(d)
1 6
1 4 [ IIT 2005 ]
1
t 2 f ( t ) dt = 1
( 30 ) If
- sin x, x
[ 0,
2 ], then f ( 1
3 ) is
sin x
(a) 3
(b) 1 3
t2
( 31 ) If
x f ( x ) dx = 0
(a)
-
2 5
2 5 t 5
(b) 0
(c) 1
(d)
for t > 0, then f
(c)
2 5
(d) 1
[ IIT IIT 2005 ]
3
4 25
is
[ IIT 2004 ]
9 - INTEGRAL CALCULUS
Page 6
( Answers at the end e nd of all questions que stions ) 1
( 32 ) 0
1
x
1
x
(a)
dx is equal to
1
2
(b)
1
2
(c) 1
(d)
( 33 ) If the area bounded by the curves x = ay 1
(a)
(b)
3 x2
1
1 3
1 2
(c)
2
[ 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
( 34 ) If f ( x ) = x2
( a ) ( 0,
)
-
(b) (
, 0)
(c) [
- 2, 2 ]
( d ) nowhere
[ IIT 2003 ]
1
( 35 ) If
I
tm ( 1
( m, n ) =
t ) n dt ,
m, n
R, then
I
( m, n ) is
0
(a) (c)
n 1
m
m
I
2n 1 m
m 1
m
1 , (n m
I
1) 1 , (n
( 36 ) Area bounded bounded by the the curves y = (a) 2
( b ) 18
3
1)
(b) 2
(c) 9
(c) 2
2
2n 1 m
(d)
m n 1
m 1 I
m
n
m
I
1 , (n
1 , (n 1)
1) [ IIT 2003 ]
x , x = 2y + 3 in the first first quadrant quadrant and and X-axis X-axis is
( 37 ) The area bounded by the curves y = (a) 1
(b)
(d)
l
xl
34 3
[ IIT 2003 ]
- 1 and y = - l x l + 1 is
(d) 4
[ IIT IIT 2002 2002 ]
x
( 38 ) If f ( x ) =
2
t 2 dt , then the real roots of the equation x
2
- 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 e nd of all questions que stions )
( 39 ) Let T > 0 be a fixed real number. Suppose Suppose f is a continuou continuous s function function such that for for T
all x
R, f ( x + T ) = f ( x ). If
I
=
3
f ( x ) dx , then the value of 0
3 2
(a)
(b)
I
I
(c)
3I
x
ln
3T
f ( 2x ) dx
is
3
(d)
6I
[ IIT 2002 ]
1 2
( 40 ) The integral equals 1
1
x
1
x
dx equals
2
(a)
-
1
(b) 0
2
(c) 1
1
( d ) 2 ln
[ IIT 2002 ]
2
x
( 41 ) If f : ( 0,
)
R, F ( x ) =
f ( t ) dt
2
and
2
F ( x ) = x ( 1 + x ),
then f ( 4 ) equals
0
(a)
5 4
(b) 7
cos 2 x
( 42 ) The value of
ax
1
(a)
(c) 4
dx,
(b) a
( 43 ) If f ( x ) =
(a ) 0
(c)
e cos x sin x 2
(b) 1
(d) 2
a > 0, is
for l x l
(c) 2
e 1
(b)
5 2
then
(c) 3
f ( x ) dx 2
(d) 3
log e x x
[ IIT 2001 ]
3
2,
otherwise,
( 44 ) The value of the integral 3 2
(d) 2
2
e2
(a)
[ IIT 2001 ]
dx
(d) 5
[ IIT 2000 ]
is
[ IIT 2000 ]
9 - INTEGRAL CALCULUS
Page 8
( Answers at the end e nd of all questions que stions ) ex ( x
( 45 ) If f ( x ) = (a) (
- , -2)
Let
g(x) =
1) ( x
2 ) dx , then then f decreases in in the the interval interval
( b ) ( - 2,
-1)
( c ) ( 1, 2 )
x ( 46 )
f ( t ) dt ,
where f is such that
0 0
f ( t ) 3 2 3 2
(a) (c)
( 47 )
1 2
for t
g( 2)
1 2
f ( t )
1
)
for
[ IIT 2000 ]
t
[ 0, 1 ]
and
[ 1, 2 ]. Then g ( 2 ) satisfies the inequality
1 2 5 2
g( 2)
( d ) ( 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 3 2
value of the integral
[ 2 sin x ] dx is 2
(a)
-
(b) 0
(c)
-
(d)
2
2
[ IIT 1999 ]
3
4 ( 48 )
dx cos x
1
=
4 (a) 2
( 49 )
(b)
(c)
1 2
(d)
-
1 2
[ IIT 1999 ]
For which of the following values of curve y = x (a)
( 50 )
-2
-4
-
m, is the area of the region bounded by the 9 2 x and the line y = mx equals ? 2
(b)
If f ( x ) = x
-2
(c) 2
(d) 4
[ IIT 1999 ]
- [ x ], for every real number x, where [ x ] is the integral part of x, then
1
f ( x ) dx
is
1
(a) 1
(b) 2
(c) 0
(d)
1 2
[ IIT 1998 ]
9 - INTEGRAL CALCULUS
Page 9
( Answers at the end e nd of all questions que stions ) x
cos 4 t dt , then g ( x +
( 51 ) If g ( x ) =
) equals
0
(a ) g(x) + g(
)
(b) g(x)
- g(
)
(c) g(x)g(
)
(d)
g( x ) g(
k
( 52 )
Let f be a positive positive function. If
=
I1
k
x f [ x ( 1
x ) ] dx
and
1 k
where
2k
(a) 2
I1
- 1 > 0, then
(b) k
I2
(c)
[ IIT 1997 ]
)
I2
=
f [ x ( 1
x ) ] dx ,
1 k
is
1
(d) 1
2
[ 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 and the line line x = 1 is (a)
5
6
(b)
6
1
(c)
5
(d) 6
6
[ IIT 1995 ]
2 ( 54 ) The value of
(a)
[ 2 sin x ] dx
5 3
(b)
2
(c)
dx
( 55 ) The value of 1
tan 3 x
(b) 1
(c)
0
(a) 0
-
where [ . ] represents the greatest integer function, function, is
5 3
(d)
-2
[ IIT 1995 ]
is
2
(d)
[ IIT 1993 ]
4
( 56 ) If f : R R be a differentiable 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 e nd of all questions que stions )
( 57 )
If f : R
R and g : R
R are continuous continuous functions, functions, then the value of the integral
2
[ f ( x )
f ( x ) ] [ g ( x )
g ( x ) ] dx
is
2
(a)
(b) 1
-1
(c)
(d) 0
[ IIT 1990 ]
2 e cos x cos 3 ( 2n
( 58 ) For any integer n, the integral
1 ) x dx
has the value
0
(a)
(b) 1
(c) 0
( d ) none of these
2
( 59 ) The value of the integral 0
(a)
( 60 )
4
(b)
2
cot x cot x
(c)
tan x
dx
[ IIT 1985 ]
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 ( d ) none of these
1
( 61 ) The value of the definite integral
(1
- 1 ) cos ( 3x + 4 ) [ IIT 1982 ]
2 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 e nd of all questions que stions )
Answers 1 b
2 a
3 d
4 d
5 d
6 b
7 b
21 a
22 c
23 d
24 b
25 b
26 c
27 d
41 c
42 c
43 c
44 b
45 c
46 b
61 d
62
63
64
65
66
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
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
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
67
68
69
70
71
72
73
74
75
76
77
78
79
80
