01 Indefinite Intergration. Module-5pdf

IndefiniteIntegration
Total No.of questions in Indefinite Integration are -
Level # 1 .................................................................................. 169
Level # 2 ................................................................................... 52
Level # 3 ................................................................................... 31
Level # 4 ................................................................................... 13
Total No. of questions ............................................................ 265

Integration of function
Questions
based on
Q.1   dx
x
2
sin
1 equals-
(A) sin x + cos x + c
(B) sin x – cos x + c
(C) cos x – sin x + c
(D) None of these
Q.2
z
4 5
2
 sin
cos
x
x
dx equals-
(A) 4 tan x – sec x + c
(B) 4 tan x + 5 sec x + c
(C) 9 tan x + c
(D) None of these
Q.3 (tan x + cot x) dx equals-
(A) log (c tan x)
(B) log (sin x + cosx) + c
(C) log (cx)
(D) None of these
Q.4
z
e e
e e
e e
e e
x x
x x
5 4
3 2
log log
log log


dx equals-
(A)
x2
2
+ c (B)
x3
3
+ c
(C)
x4
4
+ c (D) None of these
Q.5
z
1 2
1 2


cos
cos
x
x
dx equals-
(A) tan x + x + c (B) tan x – x + c
(C) sin x – x + c (D) sin x + x + c
Q.6 The value of   x)
(1 (1 + 3x) dx equal to -
(A) x – x2 – x3 + c
(B) x + x2 – x3 + c
(C) x + x2 + x3 + c
(D) None of these
LEVEL # 1
Q.7   9
4x
dx
2 dx is equal to -
(A)
6
1
tan–1 





2
3x
+ c (B)
6
1
tan–1 





3
2x
+ c
(C)
2
1
tan–1 





3
2x
+ c (D)
2
3
tan–1 





3
2x
+ c
Q.8 cos2x sin4x dx is equal to -
(A)
12
1
(cos 6x + 3 cos 2x ) + c
(B)
6
1
(cos 6x + 3 cos 2x) + c
(C) –
12
1
(cos 6x + 3 cos 2x) + c
(D) None of these
Q.9  1
–
x
2dx
2 equals -
(A)
2
1
log 




 
1
–
x
1
x
+ c (B)
2
1
log 





 1
x
1
–
x
+ c
(C) log 




 
1
–
x
1
x
+ c (D) log 





 1
x
1
–
x
+ c
Q.10
z x
x
2
x
x
b
a
)
b
(a 
dx equals-
(A) (a/b)x + 2x + c (B) (b/a)x + 2x + c
(C) (a/b)x – 2x + c (D) None of these
Q.11
zdx
x x
sin cos
2 2 equals-
(A) tan x – cot x + c (B) tan x + cot x + c
(C) cot x – tan x + c (D) None of these
Q.12
zsin
cos
x
x
1
dx equals-
(A) 2 cos (x/2) + c
(B) 2 sin(x/2) + c
(C)2 2 cos (x/2) + c
(D)–2 2 cos(x/2)+c

Q.13
z
sec x (tan x + sec x) dx equals-
(A) tan x – sec x + c
(B) sec x – tan x+ c
(C) tan x + sec x + c
(D) None of these
Q.14 The value of
z
sin cos
sin
x x
x


1 2
dx is-
(A) sin x + c (B) x + c
(C) cos x + c (D)
1
2
(sin x + cos x)
Q.15 The value of
z1
1 cos x
dx is-
(A)
1
2
cot (x/2) + c (B) –
1
2
cot (x/2) + c
(C) – cot (x/2) + c (D) – tan (x/2) + c
Q.16
z
cos sin
cos
2 2 2
2
x x
x

dx equals-
(A) cot x + c (B) sec x + c
(C) tan x + c (D) cosec x + c
Q.17
z
cosx
1
2 3
sin
sin
cos
x
x
x

F
H
G I
K
Jdx equals-
(A) sec x – cosec x + c
(B) cosec x – sec x + c
(C) sec x + cosec x + c
(D) None of these
Q.18
z
sin cos
sin cos
3 3
2 2
x x
x x

dx equals-
(A) sec x – cosec x + c
(B) sec x + cosec x + c
(C) sin x – cos x + c
(D) None of these
Q.19 cos cos
x x dx
3
z equals-
(A)
1
8
(sin 4x + 2 sin 2x) + c
(B)
1
8
(sin 4x – 2 sin 2x) + c
(C)
1
8
sin x sin 3x + c
(D) None of these
Q.20
z
2 3
5
x x
x

dx equals-
(A)
2 5
2 5
/
log /
b g
x
e
+
3 5
3 5
/
log /
b g
x
e
+ c
(B) loge (2x/5) + loge (3x/5) + c
(C) x + c
(D) None of these
Q.21 The value of
zsin
cos
2
1
x
x

dx is-
(A) x – sin x + c
(B) x + sin x + c
(C) – x – sin x + c
(D) None of these
Q.22 z
e2x+3 dx equals-
(A)
1
2
e2x+3 + c (B)
1
2
e2x+5 + c
(C)
1
3
e2x+3 + c (D)
1
2
e2x+4 – c
Q.23 z1
1


tan
tan
x
x
dx equals-
(A) log (cos x + sin x) + c
(B) log (cos x – sin x) + c
(C) log (sin x – cos x) + c
(D) None of these
Q.24 zsin cos
sin cos
4 4
2 2
x x
x x

dx equals-
(A) tan x + cot x – 2x + c
(B) tan x – cot x + 2x + c
(C) tan x – cot x – 2x + c
(D) None of these
Q.25
z2
4
x
1
x 
dx equals -
(A)
3
1
x3 –
x
1
+ c
(B) x3 –
x
1
+ c
(C)
3
1
x3 +
x
1
+ c
(D) None of these

Q.26 z
cos2x – cos2
cosx – cos


dx =
(A) 2 [sin x + x cos ] + c
(B) 2 [sin x + sin ] + c
(C) 2 [– sin x + x cos ] + c
(D) – 2 [sin x + sin ] + c
Q.27 zsin x
(1 + cosx)
2
2 dx equals-
(A) 2 tan x/2 + x + c (B) 2 tan x/2 – x + c
(C) tan x/2 – x + c (D) None of these
Q.28
z x
1
x
dx


equals-
(A) (x +1)3/2 + x3/2 + c
(B) (x +1)3/2 – x3/2 + c
(C)
2
3
[(x +1)3/2 + x3/2] + c
(D)
3
2
[(x +1)3/2 + x3/2] + c
Q.29
z dx
3x + 4 – 3x + 1
equals-
(A)
2
27
[(3x + 4)3/2 – (3x + 1)3/2] + c
(B)
9
2
[(3x + 4)3/2 + (3x +1)3/2 ]+ c
(C)
2
3
[(3x + 4)3/2 – (3x +1)3/2 ] + c
(D) None of these
Q.30
z tanx)
x
(sec
log 2
a
a 
dx equals-
(A) etan x + log sec x + c
(B) etan x + e logcos x + c
(C) tan x + log sec x + c
(D) sec x + log cos x + c
Integration by Substitution
Questions
based on
Q.31 The value of  1
–
x
x
)
x
(sec
dx
2
1
– is-
(A) – log (sec–1 x) + c
(B) log (sec–1 x) + c
(C)
 

sec 1
2
2
x
e j + c
(D) None of these
Q.32
z3
1
2
6
x
x 
dx equals-
(A) log (x6 +1) + c (B) tan–1 (x3) + c
(C) 3 tan–1 (x3) + c (D) 3 tan–1 (x3/3) + c
Q.33
zcos
sin
x
x
1
dx is equal to-
(A) – log (1+ sinx) + c
(B) log (1+ sinx) + c
(C) log (1– sinx) – c
(D) log (1– sinx) + c
Q.34 Evaluate : z
cot x cosec2 x dx.
(A) –
1
2
cot2 x + c (B)
1
2
cot2 x + c
(C) –
1
2
cos2 x – c (D) None of these
Q.35 Evalute : z 2
2
x
1
)
x
1
x
log(



dx
(A)
1
2
2
2
)
x
1
x
log(





 
 + c
(B)
2
2
)
x
1
x
log(





 
 + c
(C)
1
2 




 
 )
x
1
x
log( 2
+ c
(D) None of these
Q.36 The value of
z
tan (log )
x
x
is-
(A) log cos (log x) + c
(B) log sin (log x) + c
(C) log sec (log x) + c
(D) log cosec (log x) + c
Q.37 The value of
z
sin

1
2
2
1
x
x
e j dx is-
(A) (sin–1 x)3 + c (B)
1
3
(sin–1 x)3 + c
(C) 2 sin–1 x + c (D) None of these
Q.38
zdx
e e
x x
  equals-
(A) log (ex + e–x) + c (B) log (ex – e–x) + c
(C) tan–1 (ex) + c (D) tan–1 (e–x) + c

Q.39
za x
a x


dx is equal to-
(A) sin–1 (x/a) – a x c
2 2
 
(B) cos–1 (x/a) – a x c
2 2
 
(C) a sin–1 (x/a) – a x c
2 2
 
(D) a cos–1 (x/a) – a x c
2 2
 
Q.40
zsec
tan tan
2
2
1
x
x x 
dx is equal to -
(A) sec–1(tan x) + c (B) sec (tan–1 x) + c
(C) cosec–1(tan x) + c (D) None of these
Q.41 z
tan3 x sec2 x dx-
(A) z
(tan x)3 d (tan x) =
1
4
tan4 x + c
(B) z
(cos x)3 d (tan x) =
1
3
tan4 x + c
(C) z
(tan x)3 d (tan x) = –
1
4
tan4 x + c
(D) None of these
Q.42
z
sec cos
log tan
x ec x
x
b g
dx equals-
(A) log log cot x + c
(B) cot log x + c
(C) log (log tan x) + c
(D) tan log x + c
Q.43
z
cot (log )
x
x
dx is equal to-
(A) log (sin x) + c
(B) log (log x) + c
(C) log [sin (log x)]+c
(D) sin [log(log x)]+ c
Q.44
z
x e
x e
e x
e x
 


1 1
dx is equal to-
(A) log (xe – ex) + c
(B) e log (xe– ex) + c
(C) – log (xe– ex) + c
(D) (1/e)log(xe– ex)+c
Q.45 z
sec4 x tan x. dx is equal to-
(A)
sec4
4
x
+ c (B)
tan4
4
x
+ c
(C)
sec5
5
x
Q.46 z
x2 cos x3 dx is equal to-
(A) 1/3 sin (x3) + c (B) 3 sin (x3) + c
(C) sin (x3) + c (D) –1/3 sin (x3) + c
Q.47 Primitive of (sec/ tan2) is -
(A)
1
2
sec2  + c (B) – cot + c
(C) sin2 (/3) + c (D) – cosec + c
Q.48
z dx
x x
( )tan
1 2 1
  is equal to -
(A) log (tan–1 x) + c
(B) log (cot–1x) + c
(C) – log (tan–1x) + c
(D) None of these
Q.49
z
sin (tan )


1
2
1
x
x
dx is equal to-
(A) cos (tan–1x) + c
(B) –cos (tan–1x) + c
(C) cos (cot–1x) + c
(D) None of these
Q.50
z
x x
x
2 1 3
6
1
tan

dx is equal to-
(A)
1
3
(tan–1x3)2 + c (B)
1
6
(tan–1x)3+ c
(C)
1
6
(tan–1x)2 – c (D)
1
6
(tan–1x3)2 + c
Q.51
z
1

cos
sin
x
x x
dx equals-
(A) log (x + sin x) + c
(B) log(1+ cosx)+ c
(C) log (1–sin x) + c
(D) None of these

Q.52
z
1
2 3
2 3
   
F
H
G
I
K
J
x
x x
! !
..... dx equals-
(A) sin x + c (B) e–x + c
(C) ex + c (D) 1
Q.53 tan
z(3x – 5) sec (3x –5) dx equals-
(A) sec (3x – 5) + c (B) 1/3sec(3x–5)+ c
(C) tan (3x – 5) + c (D) None of above
Q.54
z x
tan
1
x
sec
x
tan
6
2
2

dx is equal to-
(A) tan–1(tan3x) + c (B) 3 tan–1(tan3x)+ c
(C)
3
1
tan–1(tan3x) + c (D) None of these
Q.55  dx
e
x
2
x
is equal to -
(A) 2
2
x
e + c (B)
2
x
e + c
(C) 1/2
2
x
e + c (D) x
2
x
e + c
Q.56
z dx
x p q x
( ) (tan )
1 2 2 2 1 2
  
=
(A)
1 1 2 2 1 2
q
q x p q x
log tan (tan )
 
 
L
N
M O
Q
P
+ c
(B) log [q tan–1 x + p q x c
2 2 1 2
  
(tan )
(C)
2
3
2 2 1 3 2
q
p q x c
( tan ) /
 

(D) None of the above
Q.57
z sec
tan tan
2
2
1
x
x x 
dx is equal to-
(A) tan–1 (sec x) + c
(B) sec–1 (tan x) + c
(C) cot–1 (sec x) + c
(D)cosec–1 (tan x)+c
Q.58
z
x2
ex3
cos ( ex3
) dx equals-
(A) sin ex3
+ c (B) 3 sin ex3
+ c
(C)
1
3
sin ex3
+ c (D) –
1
3
sin n ex3
+ c
Q.59
zsin
cos
p
p
x
x
2
dx equals-
(A)
tanp
x
p


1
1
+ c (B) tanp+1x + c
(C) (p+1) tanp+1 x + c (D) None of these
Q.60 z x
)
cos(e
e x
x
dx equals-
(A) 2 sin(e )
x
+ c (B) sin(e )
x
+ c
(C)
2
1
sin(e )
x
+ c (D) – sin(e )
x
+ c
Q.61
zx
x
tan2
dx equals-
(A) tan x + x + c
(B) tan x – x + c
(C) 2 (tan x – x ) + c
(D) None of these
Q.62
z sinx
1
cosx

dx is equal to-
(A) sinx
1 + c (B) sinx
1 + c
(C) 2 sinx
1 + c (D) 2 sinx
1 + c
Q.63
z x
cos
b
x
sin
a
sin2x
2
2
2
2

dx is equal to-
(A) 2
2
a
b
1

log (a2 sin2 x + b2 cos2 x ) + c
(B) 2
2
b
a
1

log (a2 sin2 x + b2 cos2 x ) + c
(C) log (a2 sin2 x – b2 cos2 x) + c
(D) None of these
Q.64
z x
logx)
1)(x
(x 2


dx equals-
(A) 3 (x + log x)3 + c
(B) (x + log x)3+ c
(C)
3
1
(x + log x)3 + c
(D) None of these

Q.65  dx
2x
sec
2x
tan3
equals-
(A) sec3 2x – 3 sec 2x + c
(B) sec3 2x + 3 sec 2x + c
(C)
6
1
[sec3 2x – 3 sec 2x] + c
(D)
6
1
[sec3 2x + 3 sec 2x ] + c
Q.66 ex (sin ex)dx equals-
(A) cos (ex) + c (B) – cos (ex) + c
(C) ex cos (ex) + c (D) None of these
Q.67
zx
logx)
(1 2

dx equals-
(A) 3 (1 + log x)3 + c
(B)
3
1
(1+ log x)3 + c
(C) (1+ log x)3 + c
(D) None of these
Q.68
z 4
2
–1
x
1
x
tan
x

dx equals-
(A)
4
1
(tan–1 x2)2 + c
(B)
2
1
(tan–1 x2)2 + c
(C) (tan–1 x2)2 + c
(D) None of these
Q.69 
3
2
secx
x dx equals-
(A)
3
1
log (sec x3 + tan x3) + c
(B) log (sec x3 + tan x3) + c
(C)
3
1
log (sec x3 – tan x3) + c
(D) None of these
Q.70  x
cos cot (sin x) dx equals-
(A) log cos (sin x) + c
(B) log sin (sin x) + c
(C) – log cos (sin x) + c
(D) – log sin ( sin x) + c
Q.71
zx
a x
dx is equal to -
(A) x
a log a + c (B) 2 x
a log a + c
(C) 2 x
a log10 a + c (D) 2 x
a loga e + c
Q.72  x
cos3
dx is equal to-
(A) cos x –
3
1
cos3 x + c
(B) sin x +
3
1
sin3 x + c
(C) sin x –
3
1
sin3 x + c
(D) cos x +
3
1
cos3 x + c
Q.73
z 1
x
x
dx
4

equals-
(A) sec–1 x 2 + c (B)
2
1
sec–1 x2 + c
(C) 2 sec–1 x2 + c (D) cosec–1 x 2 + c
Q.74 The primitive x
cos
1 is -
(A) 2 sin x/2 + c (B) 2 2 sin x/2 + c
(C) 1 + sin x/2 + c (D)
2
1
sin x/2 + c
Q.75
zx
x
4 4

z dx is equal to -
(A) sin–1
x
2
2
+ c (B) cos–1
x
2
2
+ c
(C)
1
2
sin–1
x
2
2
+ c (D)
1
2
cos–1
x
2
2
+ c
Q.76
zsin2x
1 + sin x
2
dx is equal to-
(A) log (1+ sin2 x) + c
(B)
1
2
log (1+ sin2 x) + c
(C) log sin 2 x + c
(D) tan–1 ( sin x) + c

Q.77
z
1
1
2


tan
tan
x
x
dx is equal to-
(A) –log (1 – tan x) + c
(B) log (2 + tan x) – c
(C) log (1 – tan x) – c
(D) log (1 + tan x) + c
Q.78
zx
x
3
8
1
dx equals-
(A) sin–1x4 + c (B)
1
4
sin–1 x3 + c
(C)
1
4
sin–1 x2 + c (D)
1
4
sin–1 x4 +c
Q.79
z x
sin
x
cos
6
2
dx equals-
(A)
3
1
cot3 x –
5
1
cot5 x + c
(B) –
3
1
cot3 x +
5
1
cot5 x + c
(C) –
3
1
cot3 x –
5
1
cot5 x + c
(D) None of these
Q.80
z 1
cos x. 1 – x
–1 2
dx equals-
(A) log (cos–1 x ) + x
(B) – log (cos–1x) + c
(C) – 2
1
x)
2(cos
1
 + c
(D) None of the above
Q.81
z x
tan
2
1
x
tan
2

dx is equal to-
(A)
2
1
log (cos2 x + 2 sin2 x ) + c
(B)
2
1
log (2cos2 x + sin2 x ) + c
(C)
4
1
log (cos2 x + 2 sin2 x ) + c
(D) None of these
Q.82   sin3x
2 . cos 3x dx =
(A)
9
2
( 2 + sin 3x)1/2 + c
(B)
3
2
( 2 + sin 3x)2/3 + c
(C)
3
2
(2 + sin 3x)3/2 + c
(D)
9
2
( 2 + sin 3x)3/2 + c
Q.83
z
sinx – cosx
1 – sin2x
esin x cos x dx =











 


4
3
,
4
x
If
(A) esin x + c (B) esin x – cos x + c
(C) esin x + cos x + c (D) ecos x – sin x + c
Q.84
zdx
x x x

z equals-
(A) log x + x x + c (B) 1 + x + c
(C) 4 1 + x + c (D) 2 x + x x + c
Q.85
z
10x + 10 log 10
x + 10
9 x
10 x dx is equal to-
(A) 10x – x10 +c (B) 10x + x10 +c
(C) (10x – x 10 )–1 +c (D) log (10x + x10) +c
Q.86
z dx
x logx log(logx)
.
equals-
(A) log (x logx) + c
(B) log (log x) + c
(C) x log (log x) + c
(D) log ({log(logx)} +c
Q.87
z 4
x
1
x

dx equals-
(A) tan–1(x2) + c
(B) 2 tan–1(x2) + c
(C) – tan–1(x2) + c
(D) 1/2 tan–1(x2) + c

Q.88 The value of z
(1+ tan x)3/2 sec2 x dx is-
(A)
2
5
(1+ tan x)1/2 + c
(B)
5
2
(1+ tan x)5/2 + c
(C)
2
5
(1+ tan x)5/2 + c
(D)
2
3
(1+ tan x)1/2 + c
Q.89
ztanx
x
ec
s 4
dx is equal to-
(A)
5
2
tanx (5 + tan2 x) + c
(B)
5
1
(C)
5
2
(D) None of these
Integration by Parts
Questions
based on
Q.90 

dx
xe x
is equal to-
(A) (x + 1) e–x + c
(B) (x – 1) e–x + c
(C) – (x + 1) e–x + c
(D) (1 – x) e–x + c
Q.91
z 2
–1
x
1
x
sin
x

dx is equal to-
(A) x – 2
x
1 sin–1 x + c
(B) 2
x
1 sin–1 x – x + c
(C) x + 2
x
1 sin–1 x + c
(D) None of these
Q.92 z
(log x)2 dx equals-
(A) (x log x)2 – 2x log x + 2x + c
(B) x (log x)2 – 2x log x + 2x + c
(C) x (log x)2 + 2x log x + 2x + c
(D) None of these
Q.93 The primitive of sin–1 x is-
(A) x sin–1 x – 2
x
1 + c
(B) x sin–1 x + (1/2) 2
x
1 + c
(C) x sin–1 x – (1/2) 2
x
1 + c
(D) x sin–1 x + 2
x
1 + c
Q.94
z
ex
sin cos
cos
x x
x

F
H
G I
K
J
2
dx is equal to-
(A) ex cos x + c (B) ex sec x + c
(C) ex sin x + c (D) None of these
Q.95
z
logx
x2
dx equal to -
(A) – x log ex + c (B) –
1
x
log (ex) + c
(C)
1
x
log
x
e
F
H
GI
K
J+ c (D) x log
x
e
F
H
GI
K
J+ c
Q.96
z
log(log )
x
x
dx equals-
(A) log x log
logx
e
F
H
G I
K
J+ c
(B) log (e/x2) + c
(C) log (x2/e) + c
(D) log x. log (e/x) + c
Q.97
 dx
e
x
2
x
3
is equal to -
(A)
2
1
(x2 + 1) ex2
+ c
(B)
2
1
(x2 –1) ex2
+ c
(C)
2
1
(1– x2) ex2
+ c
(D) None of these
Q.98
zcosx
–
1
sinx
–
x
dx =
(A) x cot
2
x
+ c (B) – x cot
2
x
+ c
(C) cot
2
x

Q.99 
3
2
3
3
2
x
sec
x
tan
x dx is equal to -
(A)
4
1
tan4 x3 + c (B)
8
1
tan4 x3 + c
(C)
12
1
tan4 x3 + c (D) None of these
Q.100
zx
e
x
dx equals-
(A) – x
e
1
x 
+ c (B) x
e
1
x 
+ c
(C) x
e
1
x 
+ c (D)
1
x
ex

+ c
Q.101  x log (1 + x)2 dx is equal to-
(A)
4
1
[2(x2 – 1) log (1+ x) – x2 + 2x] + c
(B)
4
1
[2(x2 – 1) log (1+ x) – x2 – 2x] + c
(C)
4
1
[2(x2 – 1) log (1+ x) – x2 – 2x] + c
(D) None of these
Q.102 The value of ex
z(cot x + log sin x) dx is-
(A) ex log sin x + c
(B) ex log cos x + c
(C) ex log tan x + c
(D) –ex log cos x + c
Q.103 sin–1(3x – 4x3) dx is equal to -
(A) x sin–1x + 1 2
 
x c
(B) x sin–1x – 1 2
 
x c
(C) 2 [x sin–1x + 1 2
 x ] + c
(D) 3 [x sin–1x + 1 2
 x ] + c
Q.104 ex[log (sec x + tan x) + sec x] dx equals-
(A) ex log sec x + c
(B) ex log tan x + c
(C) ex log (tan x + sec x) + c
(D) None of these
Q.105
z
ex











2
2
1
x
a
1
a
x
sin dx =
(A)
a
1
ex sin–1
a
x
+ c (B) aex sin–1
a
x
+ c
(C) ex sin–1
a
x
+ c (D)
2
2
x
x
a
e

+ c
Q.106   sinx)
(elogx
cosx dx equals-
(A) x sin x + cos x + (1/2) cos 2x + c
(B) x sin x – cos x + (1/4) cos 2x + c
(C) x sin x + cos x – (1/4) cos 2x + c
(D) None of these
Q.107
z 2
logx)
(1
x
log

dx equals-
(A)
logx
1
x

+ c (B)
logx
1
1

+ c
(C) –
logx
1
x

+ c (D) –
logx
1
1

+ c
Q.108  x
1
cot 1
–
dx equals-
(A) x tan–1 x +
2
1
log (1+ x2) + c
(B) x cot–1 1/x –
2
1
log (1+ x2) + c
(C) x cot–1 1/x +
2
1
log (1+ x2) + c
(D) None of these
Q.109
z
1
2
x
log (x2 + a2) dx =
(A)
1
x
log (x2 + a2 ) +
2
a
tan–1
x
a
+ c
(B) –
1
x
log (x2 + a2) +
2
a
tan–1
x
a
+ c
(C) –
1
x
log (x2 + a2) –
2
a
tan–1
x
a
+ c
(D) None of these

Q.110
z
1
2
x
sin
1
x
dx equals-
(A) x sin (1/x) + c
(B) – cot (1/x) + c
(C) cos (1/x) + c
(D) x cos (1/x) + c
Q.111 x
zsin x sec3 x dx equals-
(A)
1
2
[sec2x – tan x] + c
(B)
1
2
[x sec2 x – tan x] + c
(C)
1
2
[x sec2x + tan x] + c
(D)
1
2
[sec2 x + tan x] + c
Q.112 cos (log )
x dx is equal to
z -
(A)
2
x
cos (log x –  /4) + c
(B)
2
x
cos (log x +  /4) + c
(C)
2
x
cos (log x +  /4) + c
(D)
2
x
cos (log x –  /4) + cs
Q.113 z 2
x)
(log
1
–
x
log
dx equals-
(A)
x
logx
+ c
(B)
x
(logx)
+ c
2
(C) –
x
logx
+ c
(D) None of these
Q.114 z
sin–1








 2
x
1
2x
dx equals-
(A) x tan–1 x + log (1+ x2) +c
(B) x tan–1 x – log (1+ x)2 + c
(C) 2 x tan–1 x – log (1+ x2) + c
(D) None of these
Q.115
z
1
1
2
1

F
H
G I
K
J

F
H
G I
K
J
z x
e
x
x
dx is equal to-
(A) e + c
x–
1
x (B) e + c
x+
1
x
(C) e + c
x2 –
1
x (D) e + c
x2 +
1
x2
Integration of rational function
Questions
based on
Q.116
z dx
x x x
[(log ) log ]
2
4 1
 
=
(A)
1
2 5
log
log
log
x
x
 
 
L
N
M
M
O
Q
P
P
2 5
2 5
+ c
(B)
1
5
log
log
log
x
x
 
 
L
N
M
M
O
Q
P
P
2 5
2 5
+ c
(C)
1
2 5
log
log
log
x
x
 
 
L
N
M
M
O
Q
P
P
2 5
2 5
+ c
(D)
1
5
log
log
log
x
x
 
 
L
N
M
M
O
Q
P
P
2 5
2 5
+ c
Q.117
z3 1
2 2 3
2
x
x x

 
dx equals-
(A)
1
4
log(2x2 –2x+3) –
5
2
tan–1
2 1
5
x 
F
H
G I
K
J+c
(B)
3
4
log(2x2– 2x+ 3) +
5
2
tan–1
2 1
5
x 
F
H
G I
K
J+ c
(C)
3
4
log(2x2 –2x+ 3) +
5
2
tan–1
4 2
5
x 
F
H
G I
K
J+ c
(D) None of these
Q.118
z )
x
–
(1
2
–
x
–
x
2
3
dx =
(A) log
1
x
1
x


–
2
x
+ c
(B) log 







1
x
1
x
+
2
x2
+ c
(C) log 







1
x
1
x
+
2
x2
+ c
(D)
2
1
log 







1
x
1
x
–
2
x2
+ c

Q.119
z 2)
1)(x
(x
x
2
2


dx equals-
(A)
2
1
log 









2
x
1
x
2
2
+ c
(B)
2
1
log 









1
x
2
x
2
2
+ c
(C) log










2
x
1
x
2
2
+ c
(D) log 









1
x
2
x
2
2
+ c
Q.120 The value of
x
x
2
4
1
1


z dx equals-
(A)
1
2 2
log
x x
x x
2
2
2 1
2 1
 
 
F
H
G
I
K
J+ C
(B)
1
2 2
log
x x
x x
2
2
2 1
2 1
 
 
F
H
G
I
K
J+ C
(C)
1
2
tan–1
x
x
2
1
2

+ C
(D) None of these
Q.121 The value of
zdt
t xt
2
2 1
 
(x2 > 1) is -
(A)
1
1 2
 x
tan–1
t x
x


F
H
G
I
K
J
1 2
+ c
(B)
1
2 1
2
x 
log
t x x
t x x
  
  
F
H
G
G
I
K
J
J
2
2
1
1
+ c
(C)
1
2
log (t2 + 2xt + 1) + c
(D) None of these
Q.122
z
4 1
1
2
3
x x
x
 

dx equals-
(A) log {(x3 –1)/(x–1)} + c
(B) log { (x – 1)/ (x3 – 1) } + c
(C) log {(x3 – 1) (x – 1) } + c
(D) None of these
Q.123
zx
x x
4 2
1
 
dx equals-
(A)
1
3
tan–1
2 1
3
2
x 
F
H
G
I
K
J+ c
(B)
1
3
tan–1
2 1
3
2
x 
F
H
G
I
K
J+ c
(C)
2
3
tan–1
2 1
3
2
x 
F
H
G
I
K
J+ c
(D)
1
3
tan–1
2 1
3
2
x 
F
H
G
I
K
J+ c
Q.124 zx
x4
1

dx equals-
(A)
1
2
log
x
x
2
2
1
1


F
H
G
I
K
J+ c
(B)
1
2
log
x
x
2
2
1
1


F
H
G
I
K
J+ c
(C)
1
4
log
x
x
2
2
1
1


F
H
G
I
K
J+ c
(D)
1
4
log
x
x
2
2
1
1


F
H
G
I
K
J+ c
Q.125 If z 3
2
x
x
dx

=
x
A
+ B ln
1
x
x

+ C
(A) A =
2
1
, B = 1
(B) A = 1, B = –
2
1
(C) A = – 1, B = –1
(D) None of these
Q.126 z 4)
1)(x
(x
dx
2
2


is equal to-
(A)
3
1
tan–1 x –
3
1
tan–1 x/2 + c
(B)
3
1
tan–1 x –
6
1
tan–1 x/2 + c
(C)
3
1
tan–1 x +
3
1
tan–1 x/2 + c
(D) tan–1 x – 2 tan–1 x/2 + c

Q.127 z 3x
x
6
7x
x
2
3



dx is equal to-
(A)
2
1
x2 – 3x + 2 log x + c
(B)
2
1
x2 + 3x + 2 log x + c
(C)
2
1
x2 – 3x – 2 log x + c
(D) None of these
Q.128 z 1
x
2x
dx
2


is equal to-
(A) log 







1
x
1
2x
+ c (B) log 




 
1
2x
1
x
–
+ c
(C)
3
1
log 







1
x
1
2x
+ c (D)
3
1
log 




 
1
2x
1
x
–
+ c
Q.129
 
  1
x
x
dx
n dx is equal to-
(A) C
1
x
x
log
n
1
n
n










(B) C
x
1
x
log
n
1
n
n








 
(C) C
1
x
x
log n
n










(D) None of these
Q.130
zdx
x (x + 1)
4 is equal to-
(A)
4
1
log 






 
4
4
x
1
x
+ c
(B)
4
1
log 







1
x
x
4
4
+ c
(C)
4
1
log (x4 + 1) + c
(D) None of these
Q.131
z sinx)
sinx)(2
(1
cosx


dx equals-
(A) log 







sinx
1
sinx
2
+ c
(B) log 







sinx
2
sinx
1
+ c
(C)
2
1
log 







sinx
2
sinx
1
+ c
(D) None of these
Q.132
z 1)
x(x
dx
4

equals-
(A)
4
1
log 







1
x
x
4
4
+ c
(B)
4
1
log 






 
4
4
x
1
x
+ c
(C) log 






 
4
4
x
1
x
+ c
(D) None of these
Q.133
z 4
2x
x
1)
8)(x
(x
2
3




dx equals-
(A)
3
x3
+
2
x2
– 2x + c (B) x3 + x2 – 2x + c
(C)
3
1
(x3 + x2 – x)+ c (D) None of these
Integration of irrational function
Questions
based on
Q.134
z dx
5x – 6 – x2
equals-
(A) sin–1 (2x + 5) + c
(B) cos–1 (2x + 5) + c
(C) sin–1 (2x – 5) + c
(D) log 24
X
20
–
X
4
5
–
X
2 2

 + c
Q.135
z
2x + 3
x + 1
2 dx is equal to -
(A) 2 x + 1
2 + 3 log 1
x
x 2

 + c
(B) x + 1
2 + 3 log 1
x
x 2

 + c
(C) 2 x + 1
2 + 3 log 1
–
x
x 2
 + c
(D) None of these

Q.136
z
1
1
2
2


x
x
dx equals-
(A)
3
2
sin–1 x –
1
2
x 1 2
 x + c
(B)
3
2
sin–1 x +
1
2
x 1 2
 x + c
(C)
1
2
[sin–1 x – x 1 2
 x ] + c
(D) None of these
Q.137 zx x
2
8 7
  dx equals-
(A)
1
2
(x – 4) x x
2
8 7
 
+ 9 log [x – 4 + x x
2
8 7
  ] + c
(B)
1
2
(x – 4) x x
2
8 7
 
– 3 2 log [x – 4 + x x
2
8 7
  ]+c
(C)
1
2
(x – 4) x x
2
8 7
 
–
9
2
log [x – 4 + x x
2
8 7
  ] + c
(D) None of these
Q.138 z2 1
1
2
x
x x

 
dx equals-
(A) x x
2
1
  + c (B) 2 x x
2
1
  + c
(C)
1
2
x x
2
1
  + c (D) None of these
Q.139 z x)
–
x(1
dx
equals-
(A) sin–1(1–2x) + c
(B) log 2
x
4
–
x
4
x
2
–
1 2

 + c
(C) sin–1(2x–1) + c
(D) log x
4
–
x
4
1
–
x
2 2
 + c
Q.140 z 2
x
2x
dx
2


equals-
(A) log 15
16
x
8
–
x
4
15
1
–
x
4 2

 + c
(B) log 15
14
–
x
8
x
4
15
1
x
4 2



+ c
(C)
2
1
log 15
16
x
8
–
x
4
15
1
–
x
4 2

 + c
(D)
2
1
log 15
14
–
x
8
x
4
15
1
x
4 2



+ c
Q.141 z 2
x
–
3x
–
2
dx
is equal to-
(A) tan–1







 
17
3
2x
+ c (B) sec–1







 
17
3
2x
+ c
(C) sin–1







 
17
3
2x
+ c (D) cos–1







 
17
3
2x
+ c
Q.142 zx
x
2
3
1
dx equals-
(A)
2
3 1 3
 x + c (B) –
2
3 1 3
 x + c
(C)
1
3 1 3
 x + c (D) –
1
3 1 3
 x + c
Q.143
z
1
x
x
x


1
1
equals-
(A) log 1
–
x
x 2
 + sec–1x + c
(B) log 1
–
x
x 2
 – sec–1x + c
(C) log 1
x
x 2

 – sech–1x + c
(D) None of these

Q.144
z
x
x
x
1
1
2
2


dx =
(A)
1
2
[sin–1x2 + 1 4
 
x c
]
(B)
1
2
[sin–1x2 + 1 2
 
x c
]
(C) sin–1 x2 + 1 4
 
x c
(D) sin–1 x2 + 1 2
 
x c
Q.145
z )
( x
1
x
dx

is equal to -
(A) tan–1 x + c (B) cot–1
x + c
(C) 2 tan–1
x + c (D) 2 cot–1
x + c
Q.146
z x
x
dx

is equal to -
(A) log )
x
(1 + c (B) log )
x
(x  + c
(C) 2 log )
x
(x  + c (D) 2 log )
x
(1 + c
Integration of trigonometric function
Questions
based on
Q.147 z dx
x
1 sin
=
(A) 2 log tan
x
4 8

F
H
G I
K
J

+ c
(B) 2 log tan
x
4 8

F
H
G I
K
J

+ c
(C) 2 log sin
x
4 8

F
H
G I
K
J

– c
(D) 2 log sec
x
4 8

F
H
G I
K
J

– c
Q.148
zdx
3 + sin x
2 equals-
(A)
1
2 3
tan–1 







3
x
tan
2
+ c
(B)
1
3
tan–1








3
x
tan
+ c
(C)
1
2
tan–1 (2 tan x) + c
(D) None of these
Q.149 The value of
zsin
sin cos
x
x x

dx equals-
(A)
1
2
x +
1
2
log (sin x – cos x) + c
(B)
1
2
x –
1
2
(C) x + log (sin x + cos x) + c
(D) None of these
Q.150
z dx
a x b x
sin cos

equals-
(A)
1
2 2
a b

log tan tan
1
2
1
x
b
a

F
H
G I
K
J
L
N
M O
Q
P

+ c
(B)
1
2 2
a b

log tan tan
x
b
a

F
H
G I
K
J
L
N
M O
Q
P
1
+ c
(C)
1
2 2
a b

log tan tan
1
2
1
x
b
a

F
H
G I
K
J
L
N
M O
Q
P

+ c
(D) None of these
Q.151 z cosx
2sinx
1
dx


equals-
(A) log (1 + 2 tan x/2) + c
(B) log (1 – 2 tan x/2) + c
(C)
2
1
log (1 + 2 tan x/2) + c
(D)
2
1
log (1 – 2 tan x/2) + c
Q.152 z 2
cosx)
(sinx
2x
cos

dx is equal to-
(A) log (sin x – cos x) + c
(B) log (cos x – sin x) + c
(C) log (sin x + cos x) + c
(D) none of these
Q.153 z x
4cos
x
9sin
dx
2
2

is equal to-
(A) tan–1 





tanx
2
3
+ c
(B) tan–1 





tanx
3
2
+ c
(C) 6 tan–1 





tanx
2
3
+ c
(D)
1
6
tan–1 





tanx
2
3
+ c

Q.154
z dx
5 – 4cosx
equals-
(A)
3
2
tan–1 (3 tan x/2) + c
(B)
2
3
tan–1 (3 tan x/2) + c
(C) tan–1(3 tan x/2) + c
(D) None of these
Q.155
sin
sin
x dx
x
1
z equals-
(A) x + 2 [ 1 + tan (x/2)]–1 + c
(B) x + [ 1 + tan (x/2)]–1 + c
(C) x – 2 [ 1 + tan (x/2)]–1 + c
(D) None of these
Q.156 z
sin3 x dx is equal to-
(A)
1
3
cos3 x + cos x + c
(B)
1
3
cos3 x – cos x + c
(C)
1
3
(cos3 x + cos x) + c
(D) None of these
Q.157 z sin2x
–
1
dx
equals-
(A)
2
1
log tan 




 

4
2
x
+ c
(B)
2
1
log tan 




 

8
2
x
+ c
(C)
2
1
log tan 




 

4
2
x
+ c
(D) None of these
Q.158
z cosx)
2(1
sinx
dx


equals-
(A) log (1 + tan x ) + c
(B) log (1– tan x) + c
(C) log (2 + tan x/2) + c
(D) log (1 + tan x/2) + c
Q.159  x
cos
x
sin 3
2
dx is equal to-
(A)
3
1
sin3 x –
5
1
sin5 x + c
(B)
3
1
cos3 x –
5
1
sin5 x + c
(C)
5
1
sin3 x –
3
1
sin5 x + c
(D)
3
1
tan3 x –
5
1
sin5 x + c
Q.160
zx
x
2
1
dx is equal to -
(A)
3
2
(1 – x)3/2 (3x2 + 4x + 5)
(B)
15
1

log (1 – x) (3x2 + 4x + 8)
(C)
15
2

x
1 (3x2 + 4x + 8)
(D) None of these
Q.161
zdx
3 + sin x
2
equals-
(A)
1
2 3
tan–1 







3
x
tan
2
+ c
(B)
1
3
tan–1








3
x
tan
+ c
(C)
1
2
tan–1 (2 tan x) + c
(D) None of these
Q.162 The value of
zsin
sin cos
x
x x

dx equals-
(A)
1
2
x +
1
2
(B)
1
2
x –
1
2
(C) x + log (sin x + cos x) + c
(D) None of these

Some Integration of different expression of e
x
Questions
based on
Q.163
e
e
x
x


z 1
1
dx is equal to-
(A) log (ex + 1) + c
(B) log (ex – 1) + c
(C) 2 log (ex/2 + e–x/2) + c
(D) None of these
Q.164 ex

z 1 dx is equal to -
(A)









 1
–
e
e
log
–
1
e
2 x
–
2
x
–
x
+ c
(B)










 1
e
e
log
–
1
e
2 x
–
2
x
–
x
+ c
(C) 2 e e
x x
 
L
N
M O
Q
P
 
1 1 2
sin /
e j + c
(D) None of these
Q.165
zx
x
e
1
e


dx =
(A) log (1+ ex) – x – e–x + c
(B) log (1+ ex) + x – e–x + c
(C) log (1+ ex) – x + e–x + c
(D) log (1+ ex) + x + e–x + c
Q.166 z dx
(1 + e ) (1 – e )
x –x equals-
(A) log 







1
e
1
–
e
x
x
+ c (B) log 






 
1
e
1
e
x
x
–
+ c
(C)
1
2
log 






 
1
e
1
e
x
x
–
+ c (D)
1
2
log 







1
e
1
–
e
x
x
+c
Q.167 z x
e
1
dx

equals-
(A) log







 
x
x
e
e
1
+ c (B) log








 x
x
e
1
e
+ c
(C)
2
1
log








 x/2
x/2
e
1
e
+c (D) None of these
Q.168 z a
b + cex
dx is equal to -
(A)
a
b
log 







 x
x
ce
b
e
+ k
(B)
a
b
log







 
x
x
e
ce
b
+ k
(C) c log ( b + cex ) + k
(D) None of these
Q.169  x
2
e
–
1
dx
==
(A) log 




  1
–
e
e x
2
–
x
–
(B) log 




 1
–
e
–
e x
2
–
x
–
(C) log 




 
 1
e
e x
2
x
(D) log 




  1
–
e
e x
2
x

LEVEL # 2
Q.1 zx
x
5
12
1
dx is equal to-
(A) tan–1x6 + c (B) 2 tan–1x6 + c
(C)
1
6
tan–1x6 + c (D) None of these
Q.2 sin x
z dx is equal to-
(A) 2 (sin x – cos x ) + c
(B) 2 (sin x + cos x ) + c
(C) 2 (sin x – x cos x ) + c
(D) 2 (sin x + x cos x ) + c
Q.3 zdx
x x x
 log
is equal to-
(A) log x + log (log x) + c
(B) log log (1+ log x) + c
(C) log (1+ log x) + c
(D) None of these
Q.4 ex/2
z sin
x
2 4

F
H
G I
K
J

dx is equal to-
(A) ex/2 sin x/2 + c
(B) ex/2 cos x/2 + c
(C) 2 ex/2 sin x/2 + c
(D) 2 ex/2 cos x/2 c
Q.5 {sin(log ) cos(log )}
x x dx

z is equal to -
(A) sin (logx) + c (B) cos (log x) + c
(C) x sin (log x) + c (D) x cos (log x)+ c
Q.6 log10 x dx
z is equal to-
(A) log 10x + c
(B) x log10x + c
(C) x (log10x + log10e)+ c
(D) x(log10x – log10e) + c
Q.7 [(log ) / ]
2x x dx
z equals-
(A) x log 2x + c
(B) (log x log 2x)/2 + c
(C) (log x log 4 x) /2 + c
(D) None of these
Q.8 z x
a x
3 3

dx is equal to-
(A) sin–1
x
a
F
H
GI
K
J
3 2
/
+ c
(B)
2
3
sin–1
x
a
F
H
GI
K
J
3 2
/
+ c
(C)
3
2
sin–1
x
a
F
H
GI
K
J
3 2
/
+ c
(D)
3
2
sin–1
x
a
F
H
GI
K
J
2 3
/
+ c
Q.9 z dx
x a x b
sin( ) cos( )
 
is equal to-
(A) cos (a–b) log
sin( )
cos( )
x a
x b


+ c
(B)sec (a–b) log
sin( )
cos( )
x a
x b


+ c
(C) sin (a–b) log
cos( )
sin( )
x a
x b


+ c
(D) cosec (a–b) log
cos( )
sin( )
x a
x b


+ c
Q.10 x x dx
cos2
z is equal to-
(A)
x2
4
–
1
4
x sin 2x –
1
8
cos 2x + c
(B)
x2
4
–
1
4
x sin 2x +
1
8
cos 2x + c
(C)
x2
4
+
1
4
x sin 2x –
1
8
cos 2x + c
(D)
x2
4
+
1
4
x sin 2x +
1
8
cos 2x + c
Q.11 x x x
51 1 1
(tan cot )
 

z dx is equal to-
(A)
x52
52
(tan–1 x + cot–1 x) + c
(B)
x52
52
(tan–1 x – cot–1 x) + c
(C)
x52
52
+

2
+ c
(D) –
 x52
104
+

2
+ c

Q.12 zsin cos
sin cos
8 8
2 2
1 2
x x
x x


dx is equal to -
(A) sin 2x + c (B) –
1
2
sin 2x + c
(C)
1
2
sin 2x + c (D) – sin 2x + c
Q.13 If x = f”(t) cos t + f’(t) sin t, y = – f”(t) sin
t + f’(t) cos t, then
zdx
dt
dy
dt
F
H
GI
K
J
F
H
GI
K
J
L
N
M
M
O
Q
P
P
2 2
1 2
/
dt
is equal to-
(A) f’(t) + f”(t) + c
(B) f”(t) + f”(t) + c
(C) f(t) + f”(t) + c
(D) f’(t) – f”(t) + c
Q.14 If I = zex sin 2x dx, then for what value of
k, kI = ex (sin 2x – 2 cos 2x) + constant-
(A) 1 (B) 3 (C) 5 (D) 7
Q.15
zcos
cot tan
4 1
x
x x


dx equals-
(A) –
1
2
cos 4x + c (B) –
1
2
cos 4x + c
(C) –
1
8
cos 4x + c (D) None of these
Q.16 z
(x + 3) (x2 + 6x + 10)9 dx equals-
(A)
1
20
(x2 + 6x + 10)10 + c
(B)
1
20
(x + 3) (x2 + 6x + 10)10 + c
(C)
1
16
(x2 + 6x + 10)10 + c
(D)
1
38
(x + 3)19 +
1
2
( x+ 3) + c
Q.17 zx
x
2
2
1
1


b g
ex dx equal to -
(A)
x
x


1
1
ex + c (B)
x
x


1
1
ex + c
(C)
x
x  1
2
b g
ex + c (D)
x
x  1
ex + c
Q.18 A primitive of | x |, when x < 0, is-
(A)
1
2
x2 + c (B) –
1
2
x2 + c
(C) x + c (D) – x + c
Q.19 z
cos3 x e log ( sin x) dx is equal to -
(A)
1
4
esin x + c (B) –
1
4
sin4 x + c
(C) –
1
4
cos4 x + c (D) None of these
Q.20 ze x
tan1 1
1
2
2
 

F
H
G
I
K
J
x x
x
dx is equal to-
(A) xe c
x
tan

1
(B) x e c
x
2 1
tan

(C)
1
x e c
x
tan

1
(D) None of these
Q.21 zcos cos
cos
x x
x


3
3
1
dx is equal to-
(A)
2
3
sin–1 (cos 3/2 x) + c
(B)
3
2
sin–1 (cos 3/2 x) + c
(C)
2
3
cos–1 (cos3/2 x) + c
(D) None of these
Q.22 If z dx
x x
1 3

= a log
1 1
1 1
3
3
 
 
L
N
M
M
O
Q
P
P
x
x
+ b, then-
(A) a =
1
3
(B) a =
2
3
(C) a = –
1
3
(D) a = –
2
3
Q.23 zx x
x
tan
/


1
2
3 2
1
e j
dx equals to-
(A)
x x
x



tan 1
2
1
+ c (B)
x x
x



tan 1
2
1
+ c
(C)
tan


1
2
1
x x
x

Q.24 z
3 2
4 5
cos sin
sin cos
x x
x x


dx is equal to-
(A)
23
41
x +
2
41
log (4 sin x + 5 cos x) + c
(B)
23
41
x –
2
41
log (4 sin x + 5 cos x) + c
(C)
23
41
x –
2
41
log (4 sin x – 5 cos x) + c
(D) None of these
Q.25
zcos
sin
2 1
2 2
2
x x
x x x
 
 
dx equals-
(A) log (x2 + sin 2x + 2x) + c
(B) – log (x2 + sin 2x + 2x) + c
(C)
1
2
log (x2 + sin 2x + 2x) + c
(D) None of these
Q.26
zx
x x
2
2
1
1


e j
dx is equal to-
(A) log
x
x
2
1

F
H
G
I
K
J+ c (B) – log
x
x
2
1

F
H
G
I
K
J+ c
(C) log
x
x2
1

F
H
G I
K
J
+ c (D) – log
x
x2
1

F
H
G I
K
J+ c
Q.27
z
xn log x dx equals-
(A)
x
n
n

1
1
logx
n


R
S
T
U
V
W
1
1
+ c
(B)
x
n
n

1
1
logx
n


R
S
T
U
V
W
2
1
(C)
x
n
n

1
1
2
1
1
logx
n


R
S
T
U
V
W
+ c
(D)
x
n
n

1
1
logx
n


R
S
T
U
V
W
1
1
+ c
Q.28
z
tan–1 (sec x + tan x) dx equals-
(A)
x
2
+ c (B)
sec
sec tan
x
x x

+ c
(C)
x
4
  x
b g+ c (D) None of these
Q.29 z4 7
2
2
x
x x

 
dx equals-
(A) 2 log (x2 + x – 2) – 3 log
x
x


F
H
G I
K
J
1
2
+ c
(B) 2 log (x2 + x – 2) + 3 log
x
x


F
H
G I
K
J
1
2
+ c
(C) 3 log (x2 + x – 2) + 2 log
x
x


F
H
G I
K
J
1
2
+ c
(D) None of these
Q.30 zcos2 (ax + b) sin (ax + b) dx equals-
(A) –
cos ( )
3
3
ax b
a

+ c
(B)
cos ( )
3
3
ax b
a

+ c
(C)
sin ( )
3
3
ax b
a

+ c
(D) –
sin ( )
3
3
ax b
a

+ c
Q.31 z1 sec x dx equals-
(A) 2 sin–1( 2 sin x/2)+ c
(B) –2 sin–1( 2 sin x/2)+ c
(C) 2 log 1
–
2
x
sin
2
2
x
sin
2 2

(D) None of these
Q.32 zsin
sin
x
x
3
dx is equal to -
(A)
1
2 3
log
3
3


F
H
G
I
K
J
tan
tan
x
x
+ c
(B)
1
2 3
log
3
3


F
H
G
I
K
J
tan
tan
x
x
+ c
(C)
1
3
log
3
3


F
H
G
I
K
J
tan
tan
x
x
+ c
(D) None of these
Q.33
z1
1


cos
cos cos
x
x x
b g
dx is equal to -
(A) log ( sec x + tan x) – 2 tan x/2 + c
(B) log (sec x + tan x) + 2 tan x/2 + c
(C) log (sec x – tan x) – tan x/2 + c
(D) None of these

Q.34
z
ex
1 ex dx is equal to-
(A)
2
3
( 1+ ex )3/2 + c (B)
3
2
(1+ ex)3/2 + c
(C) ex (1+ ex)3/2 + c (D)
2
3
ex(1 + ex)3/2+ c
Q.35 If
zdx
x
5 4
 sin
= A tan
an–1 






3
4
2
x
tan
B + C,
then -
(A) B =
3
2
(B) A =
3
1
(C) A =
3
2
(D) B = –
3
5
Q.36
ze
x x
x
log /
/
1 1
2 2
2
1


e j
e j dx equals-
(A)
e
x
log /
1 1 2
e j + c
(B)
1
2
tan–1
x
x
2
1
2

F
H
G
I
K
J+ c
(C)
1
2
tan–1
x
x
2
1
2

F
H
G
I
K
J+ c
(D) tan–1
x
x
2
1
2

F
H
G
I
K
J+ c
Q.37
z
u
d v
dx
2
2
dx –
z
v
d u
dx
2
2
dx is equal to -
(A) uv + c (B) 2
du
dx
+
dv
dx
+ c
(C) u
dv
dx
– v
du
dx
+ c (D) u
dv
dx
+ v
du
dx
+ c
Q.38
z
log ( )
x x
x
 

1
1
2
2
dx equals-
(A)
1
2
[log (x + 1 2
 x ) ]2 + c
(B) log (x + 1 2
 x ) + c
(C) log ( x + 1 2
 x ) ]2 + c
(D) None of these
Q.39 If
z
f(x) dx = f (x), then
z
{f(x)}2 dx is equal
to -
(A)
1
2
{f(x)}2 (B) {f(x)]3
(C)
1
3
{f(x)]3 (D) {f(x)]2
Q.40
z
x
x x
3
3
1


dx equal to -
(A) x – log x +
2
1
log (x2 + 1) – tan–1 x + c
(B) x + log x +
1
2
log (x2 + 1) – tan–1 x + c
(C) x – log x –
2
1
log(x2 + 1) – tan–1 x + c
(D) None of these
Q.41
zx dx
x
5
3
1
equals-
(A)
2
3
( )
1 3
 x (x2 + 2) + c
(B)
2
9
( )
1 3
 x (x3 – 4) + c
(C)
2
9
( )
1 3
 x (x3 + 4) + c
(D)
2
9
( )
1 3
 x (x3 – 2) + c
Q.42
ztan cot
x x

e jdx equals-
(A) 2 tan–1
1
2
tan cot
x x

e j+ c
(B) 2 tan–1
1
2
tan cot
x x

e j+ c
(C) tan–1
1
2
tan cot
x x

e j+ c
(D) None of these
Q.43 z555x
. 55x
. 5x dx is equal to-
(A)
5
5
5
3
x
(log )
+ c (B)
5
5
5
3
5x
(log )
+ c
(C) 555x
(log 5)3 + c (D) None of these

Q.44 z
[1+ 2 tan x (tanx + sec x)]1/2 dx equals-
(A) log sec x + log (sec x + tan x) + c
(B) log sec x – (sec x + tan x) + c
(C) log (sec x + tan x) / sec x + c
(D) None of these
Q.45 z dx
x x
 
 
b g
b gdx,  

b gequals-
(A) 2 sin–1
x
c




 
(B)
1
2
sin–1
x
c




 
(C) 2 sin–1
x
c




 
(D) None of these
Q.46 zsin tan

R
S
|
T
|
U
V
|
W
|
1
2
1
x
x
dx equals-
(A) 1 2
 x + c (B)
1
2
x2 + c
(C) cos 1 2
 x + c (D) – cos 1 2
 x + c
Q.47
z x
cos
x
sin
2x
sin
4
4

dx is equal to-
(A)
2
1
tan–1(tan2x) + c
(B) 2 cot–1(tan2x) + c
(C) tan–1(tan2x) + c
(D) None of these
Q.48
ze
1 + e
x
2x dx is equal to -
(A) sin–1 (ex) + c
(B) log | (ex) + x
2
e
1 | + c
(C) cos–1(ex) + c
(D) log | (ex) + 1
–
e x
2 | + c
Q.49 zdx
1 – e2x equals-
(A) log |
e
1
e
| x
2
–
x
–

 + c
(B) log |
e
–
1
e
| x
2
–
x
–
 + c
(C) log |
e
1
e
| x
2
x

 + c
(D) log |
e
–
1
e
| x
2
x
 + c
Q.50  




 
 2
2
a
x
x
log dx is equal to-
(A) C
a
x
a
x
x
log
x 2
2
2
2











(B) C
a
x
2
a
x
x
log
x 2
2
2
2











(C) C
a
x
a
x
x
log
x 2
2
2
2











(D) None of these
Q.51 Integral of 2
)
x
(log
1
1

w.r.t. log x is-
(A) C
x
)
x
(log
tan 1


(B) tan–1 (log x) + C
(C) C
x
x
tan 1


(D) none of these
Q.52 Integral of
4
x
1
2

w.r.t. (x2 + 3) is equal to-
(A) C
4
x2


(B) C
4
x
1
2


(C) C
4
x
2 2


(D) None of these

LEVEL # 3
Q.1 If   x
x
4
1
2
dx = K sin–1 (2x) + C, then K
is equal to-
(A) log 2 (B)
2
1
log 2
(C)
2
1
(D)
2
log
1
Q.2 If  g(x) dx = g(x), then  g(x) {f(x) + f' (x)}
is equal to-
(A) g(x) f(x) – g(x) f'(x) + C
(B) g(x) f'(x) + C
(C) g(x)f(x) + C
(D) g(x) f2(x) + C.
Q.3  '
f (ax + b) {f(ax+b)}n dx is equal to
(A) 1
n
except
n
,
C
)}
b
ax
(
f
{
1
n
1 1
n







(B) n
,
C
)}
b
ax
(
f
{
)
1
n
(
1 1
n





(C) 1
n
except
n
,
C
1
n
)}
b
ax
(
f
{
)
1
n
(
a
1







(D) .
n
,
C
)}
b
ax
(
f
{
)
1
n
(
a
1 1
n





Q.4  x
cos
x
sin
1
3 dx is equal to-
(A) C
x
tan
2


(B) C
x
tan
2 
(C) C
x
tan
2
 (D) .
C
x
tan
2 

Q.5 The value of the integral  


)
1
x
(
x
x
log
)
1
x
log(
dx is-
(A) C
x
log
)
1
x
log(
)
x
(log
2
1
)]
1
x
[log(
2
1 2
2






(B) C
x
log
).
1
x
log(
]
)
x
(log
)}
1
x
[{log( 2
2






(C) C
1/x)]
[log(1
2
1 2



(D) (A) and (C) is correct
Q.6 If  log
x (1 + 1/x) dx =f(x). log (x +1) + g (x).
x2 + Ax + C, then-
(A) f(x) =
2
1
x2
(B) g (x) = log x
(C) A = 1
(D) None of these
Q.7 If  
 )
4
x
)(
1
x
(
1
2
2
dx = A tan–1
x + B tan–1
2
x
+ C, then-
(A) A = 1/3 (B) B = –1/6
(C) A = – 1/3 (D) (A) and (B)
Q.8 If  x
sin
x
cos
2
4
dx = A cot x + b sin 2 x + C
x/2 + D, then-
(A) A = – 2 (B) B = – 1/4
(C) C = – 3 (D) (B) and (C)
Q.9  


)
4
x
)(
1
x
(
3
x
2
2
2
2
dx = a log 







1
x
1
x
+
b tan–1
2
x
, then (a, b) is-
(A) (– 1/2, 1/2) (B) (1/2, 1/2)
(C) (– 1,1) (D) (1, – 1)
Q.10  


2
cos
1
)
(cos
d
is equal to-
(A) C
cos 1


 (B) C


(C) C
sin 1



(D) C
)
(cos
sin 1




Q.11 







 2
x
tan
x
)
x
cos
1
(
log dx is equal to-
(A) x tan
2
x
(B) log (1 + cos x)
(C) x log (1 + cos x)
(D) None of these
Q.12
2
2
x
x
1
x
–
1
e









 dx is equal to-
(A) C
x
1
x
1
e
2
x











(B) C
x
1
1
x
e
2
x











(C) C
x
1
1
.
e
2
x


(D) None of these
Q.13  
 )
b
x
)(
a
x
(
x
2
2
2
2
2
dx is equal to-
(A)
  C
a
x
tan
a
b
x
tan
b
a
b
1 1
1
2
2











(B) C
a
x
tan
a
b
x
tan
b
a
b
1 1
1
2
2











(C) C
a
x
tan
a
b
x
tan
b
a
b
1 1
1
2
2












(D) None of these
Q.14   )
x
1
(
x
x
–
1
7
7
dx
(A) n
 x +
7
2
n
 (1 + x7) + c
(B) n
 x
7
2
–
n
 (1 – x7) + c
(C) n
 x –
7
2
n
 (1 + x7) + c
(D) n
 x +
7
2
n
 (1 – x7) + c
Q.15 


 3
2
2
)
x
1
(
x
1
xdx
(A)
2
1
n
 (1 + 2
x
1 ) + c
(B) 2
x
1
1
2 
 + c
(C) 2 (1 + 2
x
1 ) + c
(D) None
Q.16  

x
cos
x
x
sin
x
x
cos
2
2
dx
(A) n
 





x
x
cos
+ c (B) n
 




 
x
x
cos
x
+ c
(C) n
 







 x
cos
x
x
+ c (D) None
Q.17 z dx
2e – 1
x equals-
(A) sec–1
2ex + c (B) sec–1
 
x
e
2 + c
(C) 2sec–1
 
x
e
2 + c (D) 2 sec–1 x
2e +c
Q.18 zsin
cos
3
2
3
x
x
dx equals-
(A) 3 cos x
3
1
7
1
2
cos x 
F
H
G I
K
J+ c
(B) 3 cos x
3
( 7 cos2 x – 1) + c
(C) log cos3
1
x  + c
(D) None of these
Q.19 z
cos sin
sin
x x
x

2
dx equals-
(A) log |sinx + cosx + 2
x
2
sin  | + c
(B) log |sinx + cosx + x
2
sin | + c
(C) log (sinx + cosx) + c
(D) None of these

Q.20 z
sin cos
sin cos
 
 


1 1
1 1
x x
x x
dx is equal to-
(A)
2

2 1 1
1
x x x x
  

b g b g
sin – x + c
(B)
2

2 1 1
1
x x x x
  

b g b g
sin + x + c
(C)
2

2 1 1
1
x x x x
  

b g b g
sin – x+ c
(D) None of these
Q.21
z
cos sin
sin
x x
x

2
dx is equal to-
(A) x
2
sin
x
cos
–
x
sin
log  + c
(B) sin–1(sin x – cos x) + c
(C) x
2
sin
x
cos
x
sin
log 
 + c
(D) cos–1(sin x + cos x) + c
Q.22 If
z
4 6
9 4
e e
e e
x x
x x



 dx = Ax + B log (9e2x– 4) + c,
then values of A and B are-
(A) – 19/36, – 35/36
(B) – 3/2, – 35/36
(C) –3/2,–35/36
(D) –3/2, 35/36
Asseration & Reason
(A) both S1 and S2 are true and S2 is the
correct reason of S1
(B) both S1 and S2 are true and S2 is not the
correct reason of S1
(C) S1 is true and S2 is false
(D) S1 is false and S2 is true
Q.23 Observe the following statements :
Assertion (S1) :







 

x
1
x
2
2
2
e
x
1
–
x
dx = x
1
x2
e

+ c
Reason (S2) : 
)
x
(
f
e
)
x
(
f dx = ef(x) + c.
Q.24 Assertion (S1) :   11
2
/
9
x
1
x
dx =
11
2
log
|
x
1
x
| 11
2
11

 + c
Reason (S2) :  2
x
–
1
dx
= log |x + 2
x
1 | + c
Q.25 Asseration (S1)  

10
x
e
x
9
x
10
10
log
10
x
10
= bg |10x + x10| + c
Reason (S2) :  )
n
(
f
)
x
(
'
f
dx = log |f (x)| + c
Q.26 Assertion (S1) : The indefinite integral of an odd
function is an even function.
Reason (S2) For an odd function
f (x), f (–x) = – f (x)
Q.27 Assertion (S1) : The indefinite integral of a
periodic function is periodic
Reason (S2) If f (x) is periodic with period
T  0 then f (x + T) = f (x) for all x
Q.28 Assertion (S1) :
x
1
–
x
2
2
2
e
x
1
–
x
 







dx = x
1
–
x2
e
+ c
Reason (S2) : c
)
x
(
f
dx
e
)
x
(
'
f )
x
(
f



Q.29 Assertion (S1) :  xdx
tan
x
2
tan
x
3
tan
=
3
|
x
3
sec
|
n

–
2
|
x
2
sec
|
n

– n
 |secx| + c
Reason (S2) tan 3x – tan 2x – tan x = tan 3x
tan 2x tan x
Q.30 Assertion (S1) :
 

 c
x
sin
e
dx
)
x
cos
x
(sin
e x
x
Reason (S2) : c
)
x
(
f
e
dx
)
x
(
'
f
)
x
(
f
(
e x
x




Q.31 Assertion (S1) :   2
x
–
x
2
2
x
2
–
5
dx =
2
x
–
x
2
2
2  + 3 sin–1








3
1
–
x
+ c
Reason (S2) :  2
2
x
–
a
1
dx
=
2
x 2
2
x
–
a +
2
a2
sin–1
a
x
Assertion (S1) :  

10
x
e
x
9
x
10
10
log
10
x
10
= bg |10x + x10| + c
Reason (S2) :  )
n
(
f
)
x
(
'
f
dx = log |f (x)| + c

LEVEL # 4
[PREVIOUSLY ASKED QUESTIONS IN AIEEE & IIT]
Section - A
Q.1  

1
x
2
cos
1
x
2
cos
dx =
(A) tan x – x + c
(B) x + tan x + c
(C) x – tan x + c
(D) – x – cot x + c
Q.2  2
x
)
x
(log
dx=
(A)
2
1
(log x + 1) + c
(B) –
x
1
(log x + 1) + c
(C)
x
1
(log x – 1) + c
(D) log (x + 1) + c
Q.3 If  
 )
x
sin(
x
sin
dx = Ax + B log sin (x – ) + C,
then value of (A,B) is-
(A) (sin , cos ) (B) (cos , sin )
(C) (–sin , cos ) (D) (–cos , sin )
Q.4   x
sin
x
cos
dx
is equal to-
(A)
2
1
log 




 

8
2
x
tan + C
(B)
2
1
log 





2
x
cot + C
(C)
2
1
log 




 

8
3
2
x
tan + C
(D)
2
1
log 




 

8
3
2
x
tan + C
Q.5  











2
2
)
x
(log
1
)
1
x
(log
dx is equal to -
(A)
1
)
x
(log
x
log
2

+ C (B)
1
x
x
2

+ C
(C) 2
x
x
1
xe

+ C (D)
1
)
x
(log
x
2

+ C
Q.6
dx
x x
cos sin

 3
equals
(A) C
12
2
x
tan
log
2
1






 

(B) C
12
2
x
tan
log
2
1






 

(C) C
12
2
x
tan
log 





 

(D) C
12
2
x
tan
log 





 

Q.7 The value of 2 





 

4
x
sin
dx
x
sin
is
(A) x – log | sin (x –
4

) | + c
(B) x + log | sin (x –
4

) | + c
(C) x – log | cos (x –
4

) | + c
(D) x + log | cos (x –
4

) | + c
Section - B
Q.1  )
q
x
(
)
p
x
(
)
p
x
(
dx



is equal to-
(A)
q
p
2
 q
x
p
x


+ c
(B) –
q
p
2
 p
x
q
x


+ c
(C)
)
q
x
)(
p
x
(
1


+ c
(D) None of these

Q.2  2
x
)
e
x
1
(
x
)
1
x
(


dx is equal-
(A) log 







 x
x
e
x
1
e
x
+ x
e
x
1
1

+ c
(B) log 







 x
e
x
1
x
+ x
e
x
1
1

+ c
(C) log 






 
x
x
e
x
e
x
1
+ x
e
x
1
1

+ c
(D) None of these
Q.3  x
sin
x
cos
x
sin
x
cos


(2 + 2 sin 2x) dx is equal to
(A) sin 2x + c (B) cos 2x + c
(C) tan 2x + c (D) None of these
Q.4  12
x
7
x
)
7
x
2
(
dx
2



is equal to-
(A) 2 sec–1
(2x – 7) + c
(B) sec–1
(2x – 7) + c
(C)
2
1
sec–1
(2x – 7) + 2
(D) None of these
Q.5  cos x log 





2
x
tan dx is equal to
(A) sin x log 





2
x
tan + c
(B) sin x log tan
2
x
– x + c
(C) sin x log 





2
x
tan + x + c
(D) None of these
Q.6 Let F(x) be an indefinite integral of sin2
x.
STATEMENT-1 : The function F(x) satisfies
F(x + ) = F(x) for all real x.
because
STATEMENT-2 : sin2
(x + ) = sin2
x for all
real x.
(A) Statement-1 is True, Statement-2 is True
; Statement-2 is a correct explanation
for Statement-1
(B) Statement-1, is True, Statement-2 is True;
Statement-2 is NOT a correct explanation
for Statement-1
(C) Statement-1 is True, Statement-2 is False
(D) Statement-1, False, Statement-2 is True

LEVEL # 2
ANSWER KEY
LEVEL # 1
Q.No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Ans. B B A B B B B C D D A D C B C C A A A A
Q.No. 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
Ans. A A A C A A B D B C B B B A A C B C C A
Q.No. 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Ans. A C C D A A D A B D A C B C C A B C A A
Q.No. 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80
Ans. C C B C C B B A A B D C B B C A D D C B
Q.No. 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
Ans. A D A C D D D C A C A B D B B A B B C A
Q.No. 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120
Ans. A A D C C C A B B C B D A C A A B D A A
Q.No. 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140
Ans. B C A D C B A C A B B B A C A A C B C C
Q.No. 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160
Ans. C B B A C D A A A A C C D B A B B C A C
Q.No. 161 162 163 164 165 166 167 168 169
Ans. A A C B A D B A B
Q.No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Ans. C C C C C D C B B D A B C C C A A B C A
Q.No. 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
Ans. C A B A C A D C A A A A A A C B C A A A
Q.No. 41 42 43 44 45 46 47 48 49 50 51 52
Ans. D A B A C B C B B C B C

LEVEL # 3
LEVEL # 4
Section - A
Section - B
Q.No. 1 2 3 4 5 6 7
Ans. C B B D D A B
Q.No. 1 2 3 4 5 6
Ans. B A A B B A
Q.No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Ans. D C C A D D D D A D C C A C B D D A B A
Q.No. 21 22 23 24 25 26 27 28 29 30 31
Ans. B D C C A A D A A A C

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