9 - INTEGRAL CALCULUS
Page 1
( Answers at the end of all questions ) 1
1
2
∫
2 x dx ,
I2 =
0
(c) I3 = I4
(c) 3
(d) I3 > I4
(d) 4
2
2
(b) 1:3:1
[ AIEEE 2005 ]
[ AIEEE 2005 ]
(c) 2:
( log x ) 2 + 1
xe x
+ c
(b)
+ c
2
(d)
.e
1+ x
m
log x
:2
(d) 1:1:1
[ AIEEE 2005 ]
is equal to
xa
∫
( log x - 1 ) dx 1 + ( log x ) 2
(c)
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 curv 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
w w w
1
The parabolas y = 4x and x = 4y divide the square re ion bounded by the lines x = 4, y = 4 and the coordinate axes. If S1, S2 S3 a e espectively the area of these parts numbered from top to bottom, then S1: S2 S3 is
(a)
(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)
∫
3
2 x dx , then
ce .c
(b) 2
2
(5)
2
I4 =
1
(b) I1 > I2
(a) 1:2:1
(4)
∫
2
2 x dx ,
The area enclosed between the curve y = log e ( x + e ) and the coordinate axes is (a) 1
(3)
I3 =
0
(a) I2 > I1
(2)
∫
2
3
2 x dx ,
om
If I 1 =
ra
(1)
2
[ AIEEE 2005 ]
a > 0 is
π a
(d) 2π
[ AIEEE 2005 ]
9 - INTEGRAL CALCULUS
Page 2
( Answers at the end of all questions )
∑
(b) e - 1
(a) e
(8)
If
is
sin x dx - α)
∫ sin ( x
(c) 1 - e
∫ cos x
dx - sin x
( b ) ( cos α, sin α ) ( d ) ( - cos α, sin α )
is equal to
π x - + C log tan 2 8 2
1
(a)
x 3π log tan + C 8 2 2
x log cot + C 2 2
3π x + log tan + C 8 2 2
1
(d)
[ AIEEE 2004 ]
xa 3
( 10 ) The value of
[ AIEEE 2004 ]
1
(b
m
1
(c)
[ 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
om
lim
n→∞ r =1
ce .c
(7)
r n e
ra
n
∫
1 - x 2 dx
is
.e
-2
28 3
(b)
w w
(a)
w
( 11 ) The value of I =
(a) 0
(b) 1
1 3
(c)
∫
(d)
1 3
[ AIEEE 2004 ]
π 2
∫
( sin x + cos x ) 2 1 + sin 2x
0
(c) 2
dx
is
(d) 3
[ AIEEE 2004 ]
π
π ( 12 ) If
7 3
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
f(- a)
(c) -1
(d) 1
[ AIEEE 2004 ]
(c) 3
x2
∫
(a) 3
sec 2 t dt
0
x→0
(b) 2
and
[ AIEEE 2004 ]
is
x sin x (c) 1
d) 0
m
lim
( 15 ) The value of
(d) 4
x = 3
ra
(b) 2
x = 1,
ce .c
( 14 ) The area of the region bounded by the curves y = l x - 2 l, X-axis is (a) 1
g { x ( 1 - x ) } dx ,
om
(b) -3
∫
f(- a)
I then the value of 2 is I1 (a) 2
I2 =
and
1
∫
xa
( 16 ) The value of the integral I =
x ( 1 - x )n dx
[ AIEEE 2003 ]
is
0
1 n + 1
(b)
w w
.e
(a)
( 17 ) If f ( y ) = e ,
1 n + 2
g ( y ) = y,
(c)
1 1 n + 1 n + 2
t
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 t
t
(d) F(t) = 1 - e- (1 + t) t
- (1 + t)
w
(c) F(t) = e
(b) F(t) = te-
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 ) 4
d e sin x F(x ) = , dx x
( 19 ) Let
x > 0.
3 sin x3 e dx = F ( K ) - F ( 1 ), then one of the x
∫
If
1
possible values of K is ( b ) 16
( c ) 63
( d ) 64
[ AIEEE 2003 ]
om
( a ) 15
( 20 ) Let f ( x ) be a function satisfying f ’ ( x ) = f ( x ) with f ( 0 )
e2 5 2 2
(b) e +
e2 2
-
3 2
(c) e -
e2 3 2 2
(d) e +
e2 2
+
5 2
∫ x sin x dx
xa
( b ) cos x + c ( d ) cos x - sin x + c
∫
.e
1 - os 2x dx os 2x 1
w w
( a ) tan x - x + c ( c ) x - tan x + c
w
( 23 ) The
∫
is
dx 2
2
a cos x + b 2 sin 2 x 0
( b ) π ab 2
∫e
( 24 ) The value of
( c ) 3 log ( x
[ AIEEE 2002 ]
( b ) x + tan x + c ( d ) - x - cot x + c
2
(a ) πab
4
[ AIEEE 2003 ]
[ AIEEE 2002 ]
π
alue of
( a ) log ( x
f ( x ) g ( x ) dx
alue of α is
m
= - x cos x + α, then the
( a ) sin x + c ( c ) x cos x + c
( 22 ) The value of
∫
0
ra
(a) e -
ce .c
f ( x ) + g ( x ) = x . The value of the integral
is
( 21 ) If
1
2
function that satisfies
1 and g ( x ) be a
3 log x
+ 1) +c 4
+ 1) +c
( c ) π / ab
( x 4 + 1 ) -1 dx
(b) (d)
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 ) 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
π 2
∫
( t ) dt +
cos - 1 ( t ) dt
0
(b) 1
[ AIEEE 2002 ]
cos2 x
0
(a)
om
x2
(c)
π 4
(d)
is
ce .c
log x
∫
( 25 ) The value of
π
[ AIEEE 2002 ]
b2 + 1 -
equal to x - 1
∫ [x
(b)
0
for all b > 1
x + 1
(c)
then f ( x ) is
2 + 1
(d)
x
[ AIEEE 2002 ]
1 + x2
]
3 + 3x 2 + 3x + 3 + ( x + 1 ) cos ( x + 1 ) dx =
xa
( 28 )
2
m
(a)
ra
( 27 ) If the area bounded by the X-axis, the curve y = f ( x ) and the lines x = 1, x = b is
-2
c) -1
(b) 0
(d) 1
[ IIT 2005 ]
.e
(a) 4
2
2
w w
( 29 ) Find the area between the curves y = ( x - 1 ) , y = ( x + 1 ) and y = 1 3
(a)
(b)
2 3
1
w
30
∫t
If
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 )
∫
0
(a)
1-x dx is equal to 1+ x
π + 1 2
π - 1 2
(b)
(c) 1
(d)
( 33 ) If the area bounded by the curves x = ay 1 3
(b)
3 x2 + 1
∫
( 34 ) If f ( x ) =
1 2
(c)
2
(b) (- ∞, 0)
ra
(c) [-2 2
0
[ IIT 2003 ]
n I [ ( m + 1) , n - ) ] 1+ m
(b)
2n m I [ ( m + 1) , ( n - 1 ) ] 1+ m 1+ n
(c)
2n m 1+ m 1 m
(d)
m I [ ( m + 1) , ( n - 1 ) ] n+1
xa
(a)
+ 1) , ( n - 1 ) ]
w w
.e
[(m
( 36 ) Area bounded by the curves y = (a) 2
w
( d ) nowhere
m n ∈ R, then I ( m, n ) is
m ( 1 + t )n dt ,
m
∫t
[ IIT 2004 ]
e - t dt , then the interval in which f ( x ) is increasing is
1
( 35 ) If I ( m, n ) =
2
3
2004 ]
and y = ax is 1, then a s equal to
(d) 3
x2
( a ) ( 0, ∞ )
[ II
ce .c
1
(a)
2
π
om
1
( 32 )
( b ) 18
[ 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
(c) 2
2
(d) 4
[ IIT 2002 ]
x
( 38 ) If f ( x ) =
∫
2 2 - t 2 dt , then the real roots of the equation x
- 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
( c ) 3I
I
1 2
-
1 2
(a) -
[x 1
∫
( 40 ) The integral equals
]+
( d ) 6I
1+ x ln dx equals 1- x
2
(b) 0
(c) 1
( d ) 2 ln
x
∫ f(t
1 2
dt
and
2
2
F ( x ) = x ( 1 + x ),
m
( 41 ) If f : ( 0, ∞ ) → R, F ( x ) =
[ IIT 2002 ]
ce .c
(b)
3
ra
3 I 2
(a)
is
om
0
∫ f ( 2x ) dx
[ IIT 2002 ]
then f ( 4 ) equals
0
5 4
(b) 7
π
∫
cos 2 x
.e
( 42 ) The value of
(c) 4
w w
(a) π
w
( 43 ) If
(x) =
(a ) 0
1 + ax -π
(b) aπ
dx,
(c)
π 2
(c) 2
( 44 ) The value of the integral
∫
e- 1
3 2
( d ) 2π
e cos x sin x for l x l ≤ 2, 2 otherwise,
(b) 1
(b)
5 2
[ IIT 2001 ]
a > 0, is
e2
(a)
(d) 2
xa
(a)
(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
g(x) =
0 ≤ f(t) ≤
(a)
-
(c)
( 47 )
1 2
x ∫ f ( t ) dt , 0
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
ce .c
Let
( c ) ( 1, 2 )
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
ra
( 46 )
( b ) ( - 2, - 1 )
om
( a ) ( - ∞, - 2 )
2
∫
[ 2 sin x ] dx is
m
value of the integral
π 2
(b) 0
3π
∫
π 4
dx 1 + cos x
w w
(a) 2
w
( 49 )
( 50 )
π 2
(d)
π 2
[ IIT 1999 ]
(d) -
1 2
[ IIT 1999 ]
=
.e
4 ( 48 )
(c) -
xa
(a) -π
(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 (a) -4
2
(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
(b) g(x) - g(π)
(c) g(x)g(π) k
( 52 )
∫
Let f be a positive function. If I1 = 2k - 1 > 0, then
(a) 2
(b) k
(c)
is
1 2
and I2 =
[ II
1997 ]
∫ f [ x ( 1 - x ) ] dx ,
1- k
ce .c
where
g( x ) g( π)
k
x f [ x ( 1 - x ) ] dx
1- k
I1 I2
(d)
om
(a) g(x) + g(π)
(d) 1
[ IIT 1997 ]
6 5
(b)
1
m
5 6
(c)
xa
(a)
ra
( 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 (d) 6
[ IIT 1995 ]
2π
( 54 ) The value of
∫
[ 2 sin x ] dx
where [ . ] represents the greatest integer function, is
-
5π 3
(b) -π
w w
(a)
.e
π
π 2
w
( 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 ) If f : R → R and g : R → R are continuous functions, then the value of the integral π 2
∫
is
π 2
(a) π
(b) 1
(c) -1
(d) 0 π
∫
( 58 ) For any integer n, the integral
(a) π
(b) 1
(c) 0
ra
2
∫
π 4
(b)
π 2
is
( d ) none of these
[ IIT 1983 ]
.e
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
w w w
dx
t nx
(c) π
( a ) ( x - 1 ) co ( 3x + 4 ) ( c ) sin ( 3x + 4 )
( 61 )
cot x
[ IIT 1985 ]
xa
(a)
cot x
m
0
has the value
( d ) none of hese
π ( 59 ) The value of the integral
[ IIT 1990 ]
2 e cos x cos 3 ( 2n + 1 ) x dx
0
( 60 )
om
-
[ f ( x ) + f ( - x ) ] [ g ( x ) + g ( - x ) ] dx
ce .c
( 57 )
( b ) sin ( 3x + 4 ) + 3 ( x - 1 ) cos ( 3x + 4 ) ( d ) none of these
1
he value of the definite integral
(a) -1
∫
[ IIT 1982 ]
2
( 1 + e - x ) dx is
0
(b) 2
(c) 1 + e–
1
( d ) none of these
[ IIT 1981 ]
9 - INTEGRAL CALCULUS
Page 11
( Answers at the end of all questions )
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
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
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
61 d
62
63
64
65
66
67
68
69
70
71
72
73
ra m xa .e w w w
15 c
16 c
17
18 b
19 d
20 b
35 c
36 b
37 b
38 a
39 c
40 a
54 a
55 d
56 a
57 d
58 c
59 a
60 b
74
75
76
77
78
79
80
ce .c
1 b
om
Answers
