BANSALCLASSES TARGET IIT JEE 2007
M ATH EM ATIC S NUCLEUS
QUESTION BANK ON
DEFINITE & INDEFINITE INTEGRATION
Time Limit : 5 Sitting Each of 100 Minutes duration approx. approx.
Question bank on Definite & Indefinite Integration Inte gration There are 168 questions in this question bank. Select the correct alternative : (Only one is correct) (e x 1 e 3 x ) 1 dx
Q.1
Thee val Th value ue of the the def defiinite nite integr ntegral al,,
is
1
(A)
(B)
4e 2
(C)
4e 2
1 1 1 tan (D) e e 2 2 2e 2
ln
Q.2
Thee val Th value ue of of th the def defiinite nite int integ egra rall,
cos ex
0
(A) 1
(B) 1 + (sin 1)
· 2 x e x 2 dx is 2
(C) 1 – (sin 1)
(D) (sin 1) – 1
12
Q.3
Value alue of the the defi defini nite te integ ntegra rall
( sin (3x 4x ) cos (4x 1
3
1
3
3x ) ) dx
12
(A) 0
(B) x
Q. 4
Let f (x) (x) =
2
dt 1 t4
2
7 2
(D)
2
(C) 17
(D) none of these
cot 1 (e x ) dx is equal equal to : ex
1 x 1 cot (e ) ln (e2x + 1) (A) + x + c 2 ex
1 x 1 cot (e ) ln (e2x + 1) + (B) + x + c 2 ex
1 cot 1 (ex ) 2x ln (e + 1) (C) 2 ex
1 cot 1 (e x ) 2x ln (e + 1) + (D) 2 ex
x + c
x + c
1
1 Lim Lim (1 sin 2 x ) x dx k 0 k 0
(A) 2 ln 5
Q.7
(C)
(B) 17
k
Q.6
and g be the inverse of f. Then the value of g'(0) is
(A) 1
Q.5
0
e x e x 1 e x 3
(A) 4
B ansal C lasses
(B) 1
(C) e2
(D) non existent
(B) 6
(C) 5
(D) None
dx =
Q. B. on Definite & Indefinite Integration
[2]
Q.8
1 2 3 t 2 sin 2 t dt If x satisfies the equation 2 t 2t cos 1 x – t 2 1 dt x – 2 = 0 (0 < < ), then the 0 3 value x is (A) ±
2 sin
x
Q.9
If f (x) = eg(x) and g(x) = (A) 2/17
(C) ±
(D) ± 2
sin
sin
t dt then f (2) has the value equal to : 1 t4
2
Q.10
2 sin
(B) ±
(B) 0
(C) 1
(D) cannot be determined
(C) etan sec + c
(D) etan cos + c
(C) 2/9
(D) 4/9
etan (sec – sin ) d equals :
(A) etan sin + c
(B) etan sin + c
Q.11
(x · sin2x · cos x) dx =
0
(A) 0
(B) 2/9 r 4 n
Q.12
r 3
The value of Lim n
(A)
1 35
r 1
(B)
n r 4 n
2
is equal to
1 14
(C)
1 10
1 5
(D)
b c
Q.13
f (x c)dx =
a c
b
(A) f ( x )dx
(B) f ( xc)dx
a
Let I1 =
0
sin x cos x dx ; I2 = 1 sin x. cos x
(A) I1 = I2 = I3 = I4 = 0 (C) I1 = I3 = I4 = 0 but I2 0 Q.15
b
f (x )dx
(D) f ( x2c)dx
a 2c
/2
2
(cos
a
6
x )dx ; I3 =
(sin /2
0
3
1
1 x ) dx & I4 = ln 1 dx then x 0
(B) I1 = I2 = I3 = 0 but I4 0 (D) I1 = I2 = I4 = 0 but I3 0
1 x7 dx equals : x (1 x 7 )
(A) ln x +
2 ln (1 + x7) + c 7
(B) ln x
2 ln (1 x7) + c 7
(C) ln x
2 ln (1 + x7) + c 7
(D) ln x +
2 ln (1 x7) + c 7
/2n
Q.16
(C)
a
/2
Q.14
b2c
b
0
dx = 1 tan n nx
(A) 0
B ansal C lasses
(B) 4n
n (C) 4
(D) 2n
Q. B. on Definite & Indefinite Integration
[3]
x
Q.17
f (x) = t( t 1)( t2) dt takes on its minimum value when: 0
(A) x = 0 , 1
(B) x = 1 , 2
(C) x = 0 , 2
(D) x =
3 3 3
a
Q.18
f (x) dx = a
a
a
a
(A) f ( x )f ( x ) dx (B) f ( x ) f (x )dx (C) 2 f ( x ) dx Q.19
0
0
0
(D) Zero
Let f (x) be a function satisfying f ' (x) = f (x) with f (0) = 1 and g be the function satisfyingf (x) + g (x) = x2. 1
The value of the integral f ( x )g (x ) dx is 0
(A) e – Q.20
(B) e – e2 – 3
(C)
1 (e – 3) 2
2 1 ln | x | (lnx + 2) + c 3
(D) e –
1 2 3 e – 2 2
ln | x |
x
1 ln | x | dx equals :
(A)
2 1 ln | x | (lnx 2) + c 3
(B)
(C)
1 1 ln | x | (lnx 2) + c 3
(D) 2 1 ln | x | (3 lnx 2) + c
3
Q.21
1 2 5 e – 2 2
1 2
1
2 | x 3 | | 1 x | 4 dx equals: 1 2
3 9 1 (B) (C) 2 8 4 Where {*} denotes the fractional part function. (A) 4/
Q.22
0
(A) Q.23
(D)
3 2
3x 2 .sin 1 x.cos 1 dx has the value : x x 8 2
3
(B)
24 2
3
(C)
32 2
3
(D) None
Lim
2 4 2 2 sec sec 2 · ..... sec ( n 1 ) has the value equal to 6n 6 n 6 n 6 n 3
(A)
3 3
n
B ansal C lasses
(B) 3
(C) 2
(D)
Q. B. on Definite & Indefinite Integration
2 3
[4]
3
sin 2 x sin x dx can be expressed as Q.24 Suppose that F (x) is an antiderivative of f (x) = , x > 0 then x x 1
(A) F (6) – F (2) Q.25
x + c x 4 x 1
(B)
n
(C)
x 1 + c x 4 x 1
(D)
x 1 + c x 4 x 1
2n
2n
2n
2n
1 2
(C) 2
(D) none
(B) 1
(C) 2
(D) 4
(B)
2 logx 2 logx 2 dx = n 2
4
2
(A) 0 Q.28
x + c x 4 x 1
Lim 1 cos cos 2 ..... cos (n 1) equal to (A) 1
Q.27
1 1 ( F (6) – F (2) ) (C) ( F (3) – F (1) ) (D) 2( F (6) – F (2) ) 2 2
3x 4 1 Primitive of 4 w.r.t. x is : ( x x 1) 2
(A) Q.26
(B)
If m & n are integers such that (m n) is an odd integer then the value of the definite integral
cos mx ·sin nx dx = 0
(A) 0
(B)
2n n 2 m2
(C)
2m n 2 m2
(D) none 3
Q.29
x, then y dx =
Let y={x}[x] where {x}denotes the fractional part of x & [x] denotes greatest integer
0
(A) 5/6
Q.30
If
(B) 2/3
x4 1
x x 1 2
2
dx = A ln x +
(A) A = 1 ; B = 1
(C) 1
(D) 11/6
B + c , where c is the constant of integration then : 1 x2
(B) A = 1 ; B = 1
(C) A = 1 ; B = 1
(D) A = 1 ; B = 1
(B) ln2
(C) 1 + ln 2
(D) none
Q.31
1 sin x dx = 1 cos x /2
(A) 1 ln 2 Q.32
2t dt is : x 1 x 1 4 (D) 8 f (1)
Let f : R R be a differentiable function & f (1) = 4 , then the value of ; Lim (A) f (1)
B ansal C lasses
(B) 4 f (1)
(C) 2 f (1)
f ( x )
Q. B. on Definite & Indefinite Integration
[5]
f ( x )
Q.33
t
If
2
dt = x cos x , then f ' (9)
0
(A) is equal to –
Q.34
( / 2)1 / 3 5
x ·sin x
3
1 9
(B) is equal to –
1 3
(C) is equal to
1 3
(D) is non existent
dx =
0
(A) 1 Q.35
(B) 1/2
Integral of
(C) 2
12cot x(cotx cos ecx ) w.r.t. x is :
x (A) 2 ln cos + c 2 (C)
(D) 1/3
x (B) 2 ln sin + c 2
1 x ln cos + c 2 2
(D) ln sin x ln(cosec x cot x) + c 3
Q.36
If f (x) = x + x 1 + x 2 , x R then f ( x ) dx = 0
(A) 9/2
(B) 15/2
(C) 19/2
(D) none
32 x 1 8t 2 28 t 4 dt Q.37 Number of values of x satisfying the equation , is = 3 log x 1 1 ( x 1) x
(A) 0 1
Q.38
0
(B) 1
(C) 2
tan 1 x dx = x /4
(A)
0
sin x dx x
/2
(B)
0
x dx sin x
1 (C) 2 x
Q.39 Domain of definition of the function f (x) =
0
(A) R Q.40
(D) 3
(B) R+
/2
0
x dx sin x
1 (D) 2
/ 4
0
x dx sin x
dt
is x2 t2 (C) R + {0}
If e3x cos 4x dx = e3x (A sin 4x + B cos 4x) + c then : (A) 4A = 3B (B) 2A = 3B (C) 3A = 4B
(D) R – {0}
(D) 4B + 3A = 1
b
Q.41
If f (a + b x) = f (x) , then x.f (a b x ) dx = a
(A) 0
B ansal C lasses
1 (B) 2
b
a b f ( x ) dx (C) 2 a
b
a b f ( x ) dx (D) 2 a
Q. B. on Definite & Indefinite Integration
[6]
2
4 is a 2
Q.42 The set of values of 'a' which satisfy the equation ( t log 2 a ) dt = log2 0
(A) a R
(B) a R +
(C) a < 2
3
Q.43 The value of the definite integral
2 x 5(4 x 5) dx =
2 x 5(4 x 5)
2
(A) Q.44
7 3
3 5
(B) 4 2
3 2
(C) 4 3 +
b
x 3dx
0 and x 2 dx a
a
(A) 0
(C)
7 7
2 5
3 2
2 is 3
(B) 1
(C) 2
4
x tan1 x +
4
x tan1 x +
2
2
ln (1 + x2) x + c
(B)
ln (1 + x2) + x + c
(D) y
Q.46 Variable x and y are related by equation x =
0
(D) 4
4
4
(A)
1 y
(B) y
2
1 Let f (x) = Lim h 0 h (A) equal to 0
x h
x
t 1 t
2
x tan1 x
2
2
ln (1 + x2) + x + c ln (1 + x2) x + c
d 2 y . The value of 2 is equal to 2 dx 1 t 2y
(C)
dt
x tan1 x
dt
y
Q.48
(D)
tan 1 x cot 1 x dx is equal to : 1 1 tan x cot x
(A)
Q.47
4 3
Number of ordered pair(s) of (a, b) satisfying simultaneously the system of equation b
Q.45
(D) a > 2
1 y2
(D) 4y
, then Lim x · f ( x ) is
(B) equal to
x
1 2
(C) equal to 1
(D) non existent
If the primitive of f (x) = sin x + 2x 4, has the value 3 for x = 1, then the set of x for which the primitive of f (x) vanishes is : (A) {1, 2, 3} (B) (2, 3) (C) {2} (D) {1, 2, 3, 4}
Q.49 If f & g are continuous functions in [0, a] satisfying f (x) = f (a x) & g (x) + g (a x) = 4 then a
f (x ).g(x)dx = 0
a
1 (A) f (x)dx 20
B ansal C lasses
a
a
(B) 2
0
f (x)dx
(C)
a
f (x)dx
(D) 4
0
Q. B. on Definite & Indefinite Integration
f (x)dx
0
[7]
Q.50
ln x 1x
x.
(A)
1x 2 1 x
2
2
dx equals :
2 ln x 1x
x + c
x
x 2 (C) . ln2 x 1x + 2
1 x
1 x Q.51 If f (x) = (7 x 6) 1 3 (A)
31 6
+ c
2
0 x 1 1 x 2
(B)
x x 1x 2 . ln2 2
(B)
+ c
2 1 x 2 ln x 1x + x + c
(D)
, then
f (x) dx is equal to 0
(C)
1 x 2
2
32 21 1
x
1 42
(D)
55 42
x
Q.52 The value of the definite integral e e (1 x · e x )dx is equal to 0
(A) ee
(B) ee – e
(C) ee – 1
(D) e
5 4
(D) 2
2
Q.53
1 1 sin x dx has the value equal to x x 1/ 2
(A) 0
(B)
3 4
(C)
Q.54
The value of the integral
e 2x (sin 2x + cos 2x) dx =
0
(B) 2
(A) 1
z e z
0
Q.55 The value of definite integral
(A) – Q.56
l 2
n2
(B)
1 e
l 2
n2
2z
(C) 1/2
(D) zero
(C) – ln 2
(D) ln 2
dz .
A differentiable function satisfies 3 f 2(x) f '(x) = 2x. Given f (2) = 1 then the value of f (3) is (A)
3
(B) 3 6
24
(C) 6
(D) 2
e
Q.57
For In =
(ln x)ndx, n N; which of the following holds good?
1
(A) In + (n + 1) I n + 1 = e (C) In + 1 + (n +1) In = e
B ansal C lasses
(B) In + 1 + n In = e (D) In + 1 + (n – 1) In = e
Q. B. on Definite & Indefinite Integration
[8]
1 for 0 x 1 Q.58 Let f be a continuous functions satisfying f ' (ln x) = and f (0) = 0 then f (x) can be x for x 1 defined as 1 if x 0 (A) f (x) = 1 e x if x 0 x if x 0 (C) f (x) = x e if x 0
Q.59
1 if x 0 (B) f (x) = x e 1 if x 0 x if x 0 (D) f (x) = x e 1 if x 0
Let f : R R be a differentiable function such that f (2) = 2. Then the value of Limit
f (x )
x 2
2
(A) 6 f (2)
/2
Q.60
0
(A)
Q.61
(B) 12 f (2)
2 1 a 2
Let f (x) =
n
n
Lim n
k 1
1 a 2
(B)
(C)
2 1 a
2
(D) none
1 x ln then its primitive w.r.t. x is x e x 1 (B) ln x – ex + C 2
1 (C) ln2x – x + C 2
ex (D) + C 2x
tan 1 ( x ) (C) x
tan 1 (x ) (D) x2
n , x > 0 is equal to 2 k 2 x 2
(A) x tan –1(x)
Q.63
(D) none
dx has the value : 1a 2 sin 2 x
1 (A) ex – ln x + C 2 Q.62
(C) 32 f (2)
4 t3 dt is x2
(B) tan –1(x)
sin x
2 cos 2 x sin (2x ) Let f ( x) = sin 2x 2 sin 2 x sin x
cos x then 0
cos x
(A)
/2
(B) /2
[f (x) + f (x)] dx =
0
(C) 2
(D) zero
(C) 10 7
(D) 10 9
19 Q.64 The absolute value of sinx8 is less than : 10
(A) 10 10
1 x
(B) 10 11
Q.65
The value of the integral (A)
B ansal C lasses
(cos px sin qx)2 dx where p, q are integers, is equal to :
(B) 0
(C)
(D) 2
Q. B. on Definite & Indefinite Integration
[9]
Q.66
2 Primitive of f (x) = x · 2 ln ( x 1) w.r.t. x is 2
2ln ( x 1) (A) + C 2( x 2 1)
2 ( x 2 1)2ln ( x 1) (B) + C ln 2 1
( x 2 1) ln 21 (C) + C 2(ln 2 1)
( x 2 1) ln 2 (D) + C 2(ln 2 1)
n
2
t Q.67 Lim 1 dt is equal to n n 1 0 (B) e2
(A) 0 xh
x
Limit h 0
a
2
a
=
h
(B) ln2 x
(A) 0 Q.69
(D) does not exist
n t dt n t dt 2
Q.68
(C) e2 – 1
(C)
2 n x x
(D) does not exist
Let a, b, c be nonzero real numbers such that ; 2
1
(1 +
cos8x)
(ax2 +
bx + c) dx =
0
(1 + cos8x) (ax2 + bx + c) dx , then the quadratic equation
0
ax2 + bx +
c = 0 has : (A) no root in (0, 2) (C) a double root in (0, 2)
(B) atleast one root in (0, 2) (D) none
/4
Q.70
Let In =
tann x dx , then
0
(A) A.P. Q.71
1
,
I2 I4
1 I3 I5
,
1 I4 I6
(B) G.P.
,.... are in :
(C) H.P.
(D) none
Let g (x) be an antiderivative for f (x). Then ln 1 g( x ) 2 is an antiderivative for (A)
2 f ( x ) g (x ) 1 f ( x )2
(B)
2 f ( x ) g (x ) 1 g ( x ) 2
(C)
3 32
(C)
2 f (x ) 1 f (x )2
(D) none
/4
Q.72
(cos 2x)3/2. cos x dx =
0
(A)
3 16
(B)
1
Q.73 The value of the definite integral
0
(A)
4
B ansal C lasses
(B)
1 4
2
2
3
(D)
16 2
x 2 dx 1 x 2 (1 1 x 2 ) (C)
1 4
2
3 2 16
is (D) none
Q. B. on Definite & Indefinite Integration
[10]
37
Q.74
The value of the definite integral {x}2 3(sin 2x ) dx where { x } denotes the fractional part function. 19
(A) 0
(B) 6
(C) 9
(D) can not be determined
2
Q.75 The value of the definite integral 2
(A)
(B)
Q.76 Evaluate the integral :
tan x dx , is
0
(C) 2 2
2
ln (6 x 2 )
x
(D)
2 2
dx
(A)
1 [ln (6 x 2 )]3 + C 8
(B)
1 2 [ln (6x 2 )] + C 4
(C)
1 [ln (6 x 2 )] + C 2
(D)
1 [ln (6x 2 )]4 + C 16
5 6
Q.77
1 (3 sin ) 2 1 (1 sin ) 2 d 6 2 2
(A) –
(B)
3
(C) – 2 3
2x
Q.78
dt 1 Lim Let l = Lim and m = x x x ln x t x
(A) l m = l
(D) +
3
x
ln t dt then the correct statement is 1
(B) l m = m
(C) l = m
(D) l > m
ln 3
Q.79
If f (x) =
e –x +
2
e –2x +
3
e – 3x +......
+ , then
f (x) dx =
ln 2
(A) 1
(B)
If I =
n (sin x) dx then 0
(A)
(C)
/4
/ 2
Q.80
1 2
I 2
(B)
n (sin x
1 3
(D) ln 2
cos x) dx =
/4
I 4
(C)
I 2
(D) I
n n 1 dx equals Q.81 The value of ( x r ) x k k 1 0 r 1 1
(A) n Q.82
(B) n !
(C) (n + 1) !
(D) n · n !
cos3 x cos5 x dx 2 4 sin x sin x
(A) sin x 6 tan1 (sin x) + c (C) sin x 2 (sin x)1 6 tan1 (sin x) + c
B ansal C lasses
(B) sin x 2 sin1 x + c (D) sinx 2 (sin x)1 + 5 tan1 (sin x) + c
Q. B. on Definite & Indefinite Integration
[11]
3
Q.83
0
1 2 x 2 4 x 4 x 4 x 4 dx =
(A) ln
5 2
3 2
5 2
3 2
(B) ln
x
Q.84
5 2
(C) ln
The value of the function f (x) = 1 + x +
5 2
(D) none
(ln2t + 2 lnt) dt where f (x) vanishes is :
1
(A) e1
Q.85
Limit 1 n n
(C) 2 e1
(B) 0
n 1 n 1
n n2
(B) 2 2 1
(A) 2 2
n n3
.......
(D) 1 + 2 e1
n
n has the value equal to 3 (n 1)
(C) 2
(D) 4
Q.86
Let a function h(x) be defined as h(x) = 0, for all x 0. Also
function f (x). Then the value of the definite integral
(A) equal to zero
(B) equal to 1
h(x) · f (x) dx = f (0),
for every
h' ( x) · sin x dx , is
(C) equal to – 1
(D) non existent
/4
Q.87
(tann x + tann 2 x)d(x [x]) is : ( [. ] denotes greatest integer function)
0
(A)
1
(B)
n 1
1 n 2
(C)
2
(D) none of these
n 1
1
1 Q.88 Lim (1 x ) dx is equal to 0 0 (A) 2 ln 2
(B)
4 e
(C) ln
4 e
(B) x .
x
(D) 4
Q.89 Which one of the following is TRUE. (A) x . (C)
dx
x
x ln | x | C
1 . cos x dx cos x
B ansal C lasses
tan x C
(D)
dx
x ln | x | Cx
1 . cos x dx cos x
xC
Q. B. on Definite & Indefinite Integration
[12]
Q.90
2
x2n + 1· e x dx is equal to (n N).
0
(A) n !
(B) 2 (n !)
n! 2
(C)
(D)
( n 1)! 2
0
Q.91 The true set of values of 'a' for which the inequality (B) ( , 1]
(A) [0 , 1]
(3 2x 2. 3x) dx 0 is true is:
a
(C) [0, )
(D) ( , 1] [0, )
Q.92
If (2 , 3) then number of solution of the equation cos (x + 2) dx = sin is : 0
(A) 1
(B) 2
(C) 3
(D) 4.
x2
Q.93
If x · sin x =
f (t) dt where f is continuous functions then the value of f (4) is 0
(A)
Q.94
(B) 1
2
1 2
x3 (A) 2 ( x 4x 1)1 / 2
C
x (B) ( x 2 4x 1)1 / 2
C
x2 (C) 2 ( x 4x 1)1 / 2
C
1 (D) ( x 2 4x 1)1 / 2
C
e4
x2
If the value of the integral e dx is , then the value of 1
(A) 3
Q.96
0
(A)
e4
e
(B)
n x
dx is :
e
2 e4
e
(C)
2 (e4
e)
(D) 2 e4 – 1 –
1 d 1 2 x tan 2 equals 2 dx 1 x
(B)
3 1
Q.97
(D) can not be determined
(2 x 1) dx 2 3/ 2 ( x 4 x 1)
2
Q.95
(C)
Let A =
0
et d t then 1 t
(A) Aea
B ansal C lasses
6
(C)
2
(D)
4
e t dt has the value t a 1 a 1 a
(B) Aea
(C) aea
(D) Aea
Q. B. on Definite & Indefinite Integration
[13]
/2
Q.98
sin 2 sin d is equal to :
0
(B) /4
(A) 0
Q.99
(C) /2
(D)
x2 2 dx is equal to x4 4
2 1 1 x 2 tan (A) 2 2x
C
(B)
1 2x tan 1 2 (C) 2 x 2
C
2 1 1 x 2 tan (D) 2 2x
1
Q.100 If + 2 x e (A) e1
2
x2
1
1 tan 1 ( x 2 2) C 2
C
dx = e x dx then the value of is 2
0
0
(B) e
(C) 1/2e
(D) can not be determined 1
Q.101 A quadratic polynomial P(x) satisfies the conditions, P(0) = P(1) = 0 & P(x) dx = 1. The leading 0
coefficient of the quadratic polynomial is : (A) 6 (B) 6
(C) 2
(D) 3
Q.102 Which one of the following functions is not continuous on (0,)? x
1 (B) g(x) = t sin t dt 0
(A) f(x)= cotx 1
(C) h (x) = 2 2 sin x 9
Q.103 If f (x) =
(A) f
(0+)
0
0
x
3 4
3 4
1 tan 2 x sin 2 t
for 0 < x <
0 x
2
2
2 (D) l (x) = sin( x ) , x
x
t sin t dt
x sin x ,
2
2 (B) f 4 8
=–
2
(C) f is continuous and differentiable in 0,
0, (D) f is continuous but not differentiable in 2
B ansal C lasses
Q. B. on Definite & Indefinite Integration
[14]
x2 Q.104 Consider f(x) = ; g(t) = f ( t ) dt . If g(1) = 0 then g(x) equals 1 x3
1 1 x 3 (B) n 3 2
1 3 (A) n(1 x ) 3
100
Q.105 The value of the definite integral,
x
e 0
(A)
1 (1 – e –10) 2
x2
(B) 2(1 – e –10)
1 1 x 3 (C) n 2 3
1 1 x 3 (D) n 3 3
dx is equal to
(C)
1 –10 (e – 1) 2
(D)
4 1 1 e 10 2
Q.106
[2 ex] dx where [x] denotes the greatest integer function is
0
(A) 0
(B) ln 2
(C) e2
(D) 2/e
(C) 4
(D) undefined
1
Q.107 The value of (A)
1 2
1
Q.108
dx is | x | 1 (B) 2
x ln 1
0
x dx 2 =
3 3 1 2 n l (A) 4 2 Q.109 The evaluation of (A) – 1
Q.110
1
x p p q
x
1
(C)
3 1 1 ln 4 2 54
(D)
1 27 3 ln 2 2 4
p x p 2 q 1 q x q 1 dx is x 2 p 2 q 2 x p q 1 (B)
x q x p q 1
C
(C)
x q p q
x
x p
C (D) p q C x 1 1
x 3 | x| 1 dx = a ln 2 + b then : x 2 2 | x| 1
b
3 7 3 ln 2 2 2
z
C
(A) a = 2 ; b = 1 Q.111
(B)
(B) a = 2 ; b = 0
(C) a = 3 ; b = 2
(D) a = 4 ; b = 1
b
[x] dx +
a
[ x] dx where [. ] denotes greatest integer function is equal to : a
(A) a + b
(B) b a
(C) a b
(D)
ab 2
2
Q.112 If
375 x5 (1 + x2) 4 dx = 2n then the value of n is :
0
(A) 4
B ansal C lasses
(B) 5
(C) 6
(D) 7
Q. B. on Definite & Indefinite Integration
[15]
1/ 2
Q.113
0
1 1x n dx is equal to : 1 x 2 1x 1 21 n 4 3
(A)
Q.114 If
(B)
1 2 ln 3 2
1 (C) ln2 3 4
(D) cannot be evaluated.
( x 3 2 x 2 5)e3 x dx = e3x (Ax3 + Bx2 + Cx + D) then the statement which is incorrect is
(A) C + 3D = 5 (C) C + 2B = 0 /2
Q.115 Given
0
(B) A + B + 2/3 = 0 (D) A + B + C = 0
dx 1 sin x cos x = ln 2, then the value of the def. integral.
1 (A) ln 2 2
(B)
2
ln 2
(C)
4
–
/2
0
1 ln 2 2
sin x 1 sin x cos x dx is equal to (D)
2
+ ln 2
Q.116 A function f satisfying f (sinx) = cos2 x for all x and f(1) = 1 is : x3 (A) f(x) = x + 3
1 3
x3 (C) f(x) = x 3
1 3
Q.117 For 0 < x < , 2
Q.118
(D) f(x) = x
(B)
31 sin 3sin1
x cos x
1 sin x
2
x3 3
1 3
1/2
12 1 4
2 3
ln (ecos x). d (sin x) is equal to :
(A)
3 /2
(C)
x3 (B) f(x) = 3
(D)
6 1 4
31 sin 3sin1
dx is equal to :
0
(A) 2
Q.119
x
e
(A) 2 e
/2
Q.120
0
(C) zero
(D) 2
x x dx
x
(C) e
(B) (2 + )
x
(B) e
x x C
(D) e
x
x
x 1 + C
x 2 x 1 x x x 1 C
x
dx is equal to : cos6 x sin6 x
(A) zero
B ansal C lasses
(B)
(C) /2
(D) 2
Q. B. on Definite & Indefinite Integration
[16]
Q.121 The true solution set of the inequality, (A) R 1
Q.122 If
(B) ( 1, 6) n x
x
2 2 0 dz > x 0 sin x dx is :
(C) ( 6, 1)
(D) (2, 3)
1x 2
0
5x 6x
2
dx = k
ln (1 + cos x) dx then the value of k is :
0
(A) 2
(B) 1/2
(C)
2
(D) 1/2
Q.123 Let a, b and c be positive constants. The value of 'a' in terms of 'c' if the value of integral 1
(acx a bx b 1
3
3 b 5
) dx is independent of b equals
0
3c 2
(A)
Q.124
sec
(B)
2c 3
(C)
c 3
(D)
3 2c
(sec tan ) 2 d (sec tan ) [ 2 tan (sec tan )] C (A) 2
2
(B)
(sec tan ) [2 4 tan (sec tan )] C 3
(C)
(sec tan ) [ 2 tan (sec tan )] C 3
(D)
3 (sec tan ) [ 2 tan (sec tan )] C 2
2
Q.125
1
x 2 1 dx is equal to: x 4 1
(A)
1 tan1 2 2
x Limit Q.126 x x x x 1 1 (A)
(B)
1 cot1 2 2
(C)
1 1 tan1 2 2
(D)
1 2
tan1 2
x
x1
f(t) dt is equal to :
f x 1 x1
(B) x1 f (x1)
(C) f (x1 )
(D) does not exist
Q.127 Which of the following statements could be true if, f (x) = x1/3. I 9 7/3 x + 9 28 (A) I only f (x) =
B ansal C lasses
II 9 7/3 x 2 28 (B) III only f (x) =
III
IV
3 4/3 3 x + 6 f (x) = x4/3 4 4 4 (C) II & IV only (D) I & III only f (x) =
Q. B. on Definite & Indefinite Integration
[17]
/2
Q.128 The value of the definite integral
sinx sin2x sin3x dx is equal to :
0
(A)
1 3
(B) 1
Q.129
e tan x (1 x 2 )
(A) e (C) e
2 3
(D)
1 6
2 1 1 x 2 1 2 dx (x > 0) sec 1 x cos 2 1 x
tan1 x
. tan 1 x
tan 1 x
1 . 1 x 2 C sec
C
(B)
e tan
2
(D) e x
Q.130
1 3
(C)
1 x
tan 1 x
Number of positive solution of the equation, t t
2
. tan 1 x 2
2
C 2
1 . 1 x 2 C cos ec
dt = 2 (x 1) where { } denotes the fractional
0
part function is : (A) one
(B) two
(C) three
(D) more than three
1
Q.131 If f (x) = cos(tan –1x) then the value of the integral x f ' ' ( x ) dx is 0
(A)
Q.132 If
3 2 2
(B)
1 sin
(B) 1
xn
(2
0
Vn =
2 4 2 1 x 1 ( x 3x 1) tan x
1 x
B ansal C lasses
1 2
xn (1 x) n dx n
3 2 2
(D) 4 2
N, which of the following statement(s)
0
(B) Un = 2 n Vn
( x 2 1) dx
(A) ln x
(C)
2 1
x) n dx;
is/are ture? (A) Un = 2n Vn
Q.134
(D) 1
(C) 1
x x dx = A sin then value of A is: 2 4 4
(A) 2 2 Q.133 For Un =
3 2 2
(C) Un = 22n Vn
(D) Un = 2 2n Vn
= ln | f (x) | + C then f (x) is
1 x
(B) tan –1 x
1 x
(C) cot –1 x
1 x
(D) ln tan 1 x
Q. B. on Definite & Indefinite Integration
[18]
/ 3
Q.135 Let f (x) be integrable over (a, b) , b > a > 0. If I1 = / 3
I2 =
/6
I1 : I2
f (tan + cot ). cosec2 d , then the ratio
/6
(A) is a positive integer (C) is an irrational number sin x
Q.136 f (x) =
f (tan + cot ). sec2 d &
(B) is a negative integer (D) cannot be determined.
(1 t + 2 t3) d t has in [ 0, 2 ]
cos x
(A) a maximum at & a minimum at 4
(C) a maximum at
3 4
(B) a maximum at
5 7 & a minimum at 4 4
x3
Q.137 Let S (x) =
ln t d t (x > 0) and H (x) =
x2
3 7 & a minimum at 4 4
(D) neither a maxima nor minima S (x) . Then H(x) is : x
(A) continuous but not derivable in its domain (B) derivable and continuous in its domain (C) neither derivable nor continuous in its domain (D) derivable but not continuous in its domain.
d Q.138 Number of solution of the equation dx (A) 4
(B) 3
sin x
cos x
dt = 2 2 in [0, ] is 1 t2 (C) 2 (D) 0
2 sin 2 x 1 cos x ( 2 sin x 1) Q.139 Let f (x) = + then 1 sin x cos x
e
x
f ( x ) f ' ( x ) dx (where c is the constant of integeration)
(A) ex tanx + c
(B) excotx + c
(C) ex cosec2x + c
(D) exsec2x + c
x3
Q.140 The value of x that maximises the value of the integral
t(5 t) dt is x
(A) 2
(B) 0
(C) 1
(D) none
Q.141 For a sufficiently large value of n the sum of the square roots of the first n positive integers i.e. 1 2 3 ...................... n is approximately equal to (A)
1 3/ 2 n 3
(B)
2 3/ 2 n 3
(C)
1 1/ 3 n 3
2 1/ 3 (D) n 3
2
dx 2 is ( 1 x ) 0
Q.142 The value of (A) –2
B ansal C lasses
(B) 0
(C) 15
(D) indeterminate
Q. B. on Definite & Indefinite Integration
[19]
a
Q.143 If
dx xa x
0
(A)
/8
2 tan d , then the value of 'a' is equal to (a > 0) sin 2
0
3 4
(B)
Q.144 The value of the integral
(C)
4
3 4
(D)
9 16
sin ln ( 2 2x ) dx is x 1
(B) ln sin
(A) – cos ln (2x + 2) + C
2 + C x 1
+ C x 1 2
2 + C x 1
(C) cos
(D) sin 1
x 1 Q.145 If f(x) = A sin + B , f = 2 and f(x) dx = 2 A , Then the constants A and B are 2 2 0 respectively. 4 (A) & (B) 2 & 3 (C) 0 & 4 (D) & 0 2 2 2
Q.146 Let I1 =
2 e x sin(x )dx
0
2
; I2 =
2 e x dx
0
and consider the statements I I1 < I2 Which of the following is(are) true? (A) I only (C) Neither I nor II nor III
II
2
; I3 =
2
e x (1 x ) dx
0
I2 < I3
I1 = I3
III
(B) II only (D) Both I and II
2 sin x Q.147 Let f (x) = , then f (x ) f x dx = x 2 0
(A)
2
0
f ( x ) dx
(B) f ( x ) dx 0
1
Q.148 Let u =
0
ln ( x 1)
x2 1
(A) u = 4v Q.149 If f x (A)
B ansal C lasses
(D)
0
f ( x ) dx
2 dx and v = ln (sin 2 x ) dx then
0
sin x ·sin 1 cos 2 2 /16
f ( x ) dx 0
(B) 4u + v = 0 x2
(C)
1
(C) u + 4v = 0
(D) 2u + v = 0
.d then the value of f ' , is 2
(B) –
(C) 2
(D) 0
Q. B. on Definite & Indefinite Integration
[20]
2
Q.150 The value of the definite integral,
0
(A) 0
(B)
sin 5x dx is sin x
(C)
2
(D) 2
Select the correct alternatives : (More than one are correct) b
Q.151
sgn x dx = (where a, b R) a
(A) | b | – | a |
Q.152
(B) (b–a) sgn (b–a)
(C) b sgnb – a sgna
x dx = tan1 m tan + C then : 2 5 4 cos x (A) = 2/3 (B) m = 3 (C)
(D) | a | – | b |
= 1/3
(D) m = 2/3
Q.153 Which of the following are true ? a
a
x . f (sin x) dx = 2 .
(A)
a
n 0
f (sin x ) dx
f cos x dx = n. f cos x dx
2
a
dx = 2.
1
2x 2 3x 3
0
(x 1) x 2 2x 2
f (x) b
0
c
(D)
dx is :
1 3
(B) + 2 ln2 tan1
(C) 2 ln2 cot 1 3
(D) + ln4 + cot 1 2
4
4
4
cos 2 x cosec2 x dx is equal to : 1 x2
(A) cot x cot 1 x + c (C)
dx
f (x c) dx = f (x) dx
(A) + 2 ln2 tan1 2
x2
2
0
b c
0
2
a
2
Q.154 The value of
Q.155
f (x)
(B)
a
(C)
a
tan 1 x
(B) c cot x + cot 1 x
cos ec x sec x
+ c
(D)
e n tan
1 x
cot x + c
where 'c' is constant of integration . x
Q.156 Let f (x) = 0
sint dt (x > 0) then f (x) has : t
(A) Maxima if x = n where n = 1, 3, 5,..... (B) Minima if x = n where n = 2, 4, 6,...... (C) Maxima if x = n where n = 2, 4, 6,...... (D) The function is monotonic
B ansal C lasses
Q. B. on Definite & Indefinite Integration
[21]
1
dx
Q.157 If In =
1 x2
0
n
; n N, then which of the following statements hold good ?
(A) 2n In + 1 = 2 n + (2n 1) In
(C) I2 = Q.158
8
z 1
x2 1
(A)
1 4
n
(B) I2 = (D) I3 =
Q.159 If An =
0
0
sin (2 n 1) x d x ; Bn = sin x
/ 2
16
5 48
0
2
sin n x d x ; for n N , then : sin x (B) Bn + 1 = Bn (D) Bn + 1 Bn = An + 1
(B)
4
(C) is same as
0
4
x d x : (1 x) (1 x 2 )
(A)
Q.161
x 1 dx equals : x 1
(A) An + 1 = An (C) An + 1 An = Bn + 1 Q.160
8
1 2 x 1 1 2 x 1 1 2 x1 1 2 x 1 ln ln ln ln + c (B) + c (C) + c (D) + c x 1 x 1 x1 x 1 2 4 2 4 / 2
dx (1 x) (1 x 2 )
2
(D) cannot be evaluated
1 cscx dx equals
(A) 2 sin 1 sinx + c (C) c 2 sin 1 (1 2 sin x) /2
Q.162 If f (x) =
n
(1 x sin 2 ) d , x 0 then : sin 2
t 1 1
0
(A) f (t) =
(B) 2 cos 1 cosx + c (D) cos 1 (1 2 sin x) + c
(B) f (t) =
(C) f (x) cannot be determined
2 t 1
(D) none of these.
Q.163 If a, b, c R and satisfy 3 a + 5 b + 15 c = 0 , the equation ax4 + b x2 + c = 0 has : (A) atleast one root in ( 1, 0) (B) atleast one root in (0, 1) (C) atleast two roots in ( 1, 1) (D) no root in ( 1, 1)
dx x 2 dx Q.164 Let u = &v= then : 4 2 4 2 0 x 7 x 1 0 x 7 x 1
(A) v > u
B ansal C lasses
(B) 6 v =
(C) 3u + 2v = 5/6
(D) u + v = /3
Q. B. on Definite & Indefinite Integration
[22]
Q.165 If eu . sin 2x dx can be found in terms of known functions of x then u can be : (A) x (B) sin x (C) cos x (D) cos 2x x
Q.166 If f(x) =
1
n
t dt where x > 0 then the value(s) of x satisfying the equation, 1 t
f(x) + f(1/x) = 2 is : (A) 2 (B) e
(C) e 2
(D) e2 1
Q.167 A polynomial function f(x) satisfying the conditions f(x) = [f (x)]2 &
f(x) dx =
0
x2 (A) 4
3 9 x 2 4
x2 (B) 4
3 9 x 2 4
x2 (C) 4
x + 1
19 can be: 12
x2 (D) +x+1 4
Q.168 A continuous and differentiable function 'f ' satisfies the condition , x
f (t) d t = f 2 (x) 1 for all real ' x '. Then :
0
(A) ' f ' is monotonic increasing x R (B) ' f ' is monotonic decreasing x R (C) 'f ' is non monotonic (D) the graph of y = f (x) is a straight line.
B ansal C lasses
Q. B. on Definite & Indefinite Integration
[23]
ANSWER KEY
68 A ,D Q .1 B,D Q.167 C ,D Q .166
A ,B,C,D Q .165
B ,C,D Q .164
A ,B,C Q .163
A,B Q.162
A,D Q.161
A,C Q.160
A,D Q.159
B,D Q.158
A,B Q.157
A,B Q.156
B,C,D Q.155
A,C ,D Q.15 4
A,B ,C ,D Q.15 3
A,B Q.15 2
A,C Q.15 1
B
Q.150
A
Q.149
B
Q.148
A
Q.147
D
Q.146
D
45 Q. 1
A
44 Q. 1
D
43 Q. 1
D
42 Q. 1
B
41 Q. 1
C
Q.140
A
Q.139
C
Q.138
B
Q.137
B
Q.136
A
Q .135
B
Q .134
C
Q .133
D
Q .132
D
Q .131
B
Q.130
C
Q.129
D
Q.128
D
Q.127
B
Q.126
B
Q.125
C
Q.124
A
Q.123
B
Q.122
D
Q.121
B
Q.12 0
A
Q.119
D
Q.118
A
Q.117
C
Q.116
C
Q.114
A
Q.113
B
C
Q.111
C
.115 Q
.112 Q
B
0 Q.1 1
C
9 Q.1 0
A
8 Q.1 0
C
7 Q.1 0
B
6 Q.1 0
D
Q.105
B
Q.104
C
Q.103
D
Q.102
B
Q.101
A
100 Q .
D
99 Q .
B
98 Q .
B
97 Q .
A
96 Q .
B
Q.95
B
Q.94
A
Q.93
B
Q.92
D
Q.91
C
Q .90
B
Q .89
B
Q .88
A
Q .87
C
Q .86
C
Q.85
D
Q.84
C
Q.83
C
Q.82
D
Q.81
A
Q.80
B
Q.79
A
Q.78
B
Q.77
B
Q.76
B
Q.75
B
Q.74
C
Q.73
C
Q.72
B
Q.71
A
Q.70
B
Q.69
B
Q.68
C
Q.67
C
Q.66
D
5 Q .6
C
4 Q .6
A
3 Q .6
C
2 Q .6
C
1 Q .6
A
Q.60
C
Q.59
D
Q.58
C
Q.57
B
Q.56
A
Q .55
C
Q .54
A
Q .53
A
Q .52
D
Q .51
A
Q.50
B
Q.49
C
Q.48
D
Q.47
B
Q.46
D
Q.45
B
Q.44
D
Q.43
B
Q.42
C
Q.41
C
Q.40
D
Q.39
C
Q.38
B
Q.37
C
Q.36
B
Q.35
D
Q.34
A
Q.33
D
Q.32
A
Q.31
C
Q.3 0
D
Q.2 9
B
Q.2 8
A
Q.2 7
A
Q.2 6
B
Q.25
A
Q.24
A
Q.23
C
Q.22
C
Q.21
A
20 Q .
D
19 Q .
A
18 Q .
C
17 Q .
B
16 Q .
C
Q.15
C
Q.14
A
Q.13
C
Q.12
D
Q.11
D C
Q .10 Q .5
A C
Q .9 Q .4
D Q .8 B 3 Q .
A C
Q .7 Q .2
C A
Q .6 Q .1
B ansal C lasses
Q. B. on Definite & Indefinite Integration
[24]
