Integrals Test-3 Class 12 Notes | EduRev

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Class 12 : Integrals Test-3 Class 12 Notes | EduRev

 Page 1


CBSE TEST PAPER-03 
CLASS - XII MATHEMATICS (Calculus: Integrals) 
Topic: - Integrals (ANSWERS) 
1. Put 
10
10
x
x t + =
9
(10 10 .log 10 )
x
e
x dx dt
dt
t
+ =
=
?
10
log | |
log( 10 )
x
t c
x c
= +
= + +
2. 
2 2 2
2 2
cos 2 2sin 1 2sin 2sin
 = 
cos cos
x x x x
dx dx
x x
+ - +
=
? ?
2
2
1
sec 
cos
tan
dx x dx
x
x c
= =
= +
? ?
3. Let 
2
( ) sin f x x =
2
2
( ) sin ( )
sin
f x x
x
- = - =
( ) f x =
? function is even 
4 4
2 2
4 0
sin 2 sin 
1 cos 2
2
2
x dx x dx
x
dx
p p
p - ? =
- ? ?
=
? ?
? ?
? ?
?
4
sin 2
2
1
4 2
o
x
x
p
p
? ?
= - ? ?
? ?
= -
Page 2


CBSE TEST PAPER-03 
CLASS - XII MATHEMATICS (Calculus: Integrals) 
Topic: - Integrals (ANSWERS) 
1. Put 
10
10
x
x t + =
9
(10 10 .log 10 )
x
e
x dx dt
dt
t
+ =
=
?
10
log | |
log( 10 )
x
t c
x c
= +
= + +
2. 
2 2 2
2 2
cos 2 2sin 1 2sin 2sin
 = 
cos cos
x x x x
dx dx
x x
+ - +
=
? ?
2
2
1
sec 
cos
tan
dx x dx
x
x c
= =
= +
? ?
3. Let 
2
( ) sin f x x =
2
2
( ) sin ( )
sin
f x x
x
- = - =
( ) f x =
? function is even 
4 4
2 2
4 0
sin 2 sin 
1 cos 2
2
2
x dx x dx
x
dx
p p
p - ? =
- ? ?
=
? ?
? ?
? ?
?
4
sin 2
2
1
4 2
o
x
x
p
p
? ?
= - ? ?
? ?
= - 4. 
5 4
log
3 2
 
x x
dx e
x x
?
?
? ? - ? =
? ?
?
- ? ?
?
? 
( )
4
1 x x - =
( )
2
1 x x - dx
?
2
x dx =
?
3
3
x
c = +
5. 
x
e
=
( )
x x
x
e e
e
- - ( )
x x
dx
e e
- +
?
( )
 
x x
x x
put e e t
e e dx dt
- - + =
- =
( )
log | |
log
x x
dt
t c
t
e e c
- = = +
= + +
?
6. 
3 3
6 6
cos
(1)
sin cos sin
1
cos
dx x
I dx
x x x
x
p p
p p
= = - - - - - - - +
+
? ?
 
3
6
cos
3 6
cos sin
3 6 3 6
x
I dx
x x
p
p
p p
p p p p
? ?
+ - ? ?
? ?
=
? ? ? ?
+ - + + - ? ? ? ?
? ? ? ?
?
( )
3
6
( )
sin
(2)
sin sin
b b
a a
f x dx f a b x dx
x
I dx
x x
p
p
?
= + - ?
?
= - - - - - - - +
? ?
?
?
3
6
(1) (2)
2 1 I dx
p
p
+
=
?
[ ]
3
6
3 6 6
x
p
p
p p p
=
= - =
 
12
I
p
=
Page 3


CBSE TEST PAPER-03 
CLASS - XII MATHEMATICS (Calculus: Integrals) 
Topic: - Integrals (ANSWERS) 
1. Put 
10
10
x
x t + =
9
(10 10 .log 10 )
x
e
x dx dt
dt
t
+ =
=
?
10
log | |
log( 10 )
x
t c
x c
= +
= + +
2. 
2 2 2
2 2
cos 2 2sin 1 2sin 2sin
 = 
cos cos
x x x x
dx dx
x x
+ - +
=
? ?
2
2
1
sec 
cos
tan
dx x dx
x
x c
= =
= +
? ?
3. Let 
2
( ) sin f x x =
2
2
( ) sin ( )
sin
f x x
x
- = - =
( ) f x =
? function is even 
4 4
2 2
4 0
sin 2 sin 
1 cos 2
2
2
x dx x dx
x
dx
p p
p - ? =
- ? ?
=
? ?
? ?
? ?
?
4
sin 2
2
1
4 2
o
x
x
p
p
? ?
= - ? ?
? ?
= - 4. 
5 4
log
3 2
 
x x
dx e
x x
?
?
? ? - ? =
? ?
?
- ? ?
?
? 
( )
4
1 x x - =
( )
2
1 x x - dx
?
2
x dx =
?
3
3
x
c = +
5. 
x
e
=
( )
x x
x
e e
e
- - ( )
x x
dx
e e
- +
?
( )
 
x x
x x
put e e t
e e dx dt
- - + =
- =
( )
log | |
log
x x
dt
t c
t
e e c
- = = +
= + +
?
6. 
3 3
6 6
cos
(1)
sin cos sin
1
cos
dx x
I dx
x x x
x
p p
p p
= = - - - - - - - +
+
? ?
 
3
6
cos
3 6
cos sin
3 6 3 6
x
I dx
x x
p
p
p p
p p p p
? ?
+ - ? ?
? ?
=
? ? ? ?
+ - + + - ? ? ? ?
? ? ? ?
?
( )
3
6
( )
sin
(2)
sin sin
b b
a a
f x dx f a b x dx
x
I dx
x x
p
p
?
= + - ?
?
= - - - - - - - +
? ?
?
?
3
6
(1) (2)
2 1 I dx
p
p
+
=
?
[ ]
3
6
3 6 6
x
p
p
p p p
=
= - =
 
12
I
p
=
7. 
4
4 4
sin .cos
cos sin
o
x x
I dx
x x
p
=
+
?
Dividing N and D by cos
4
x 
4 4
4 4
4 4
sin .cos
cos
cos sin
cos cos
o
x x
x
dx
x x
x x
p
=
+
?
2
4
4
2
4
2 2
tan .sec
1 tan
tan .sec
1 (tan )
o
o
x x
dx
x
x x
dx
x
p
p
=
+
=
+
?
?
2
2
 tan
2 tan .sec 
put x t
x x dx dt
=
=
1
2
1
1
1
2 1
1
tan
2
o
o
dt
t
t
- =
+
? ? =
? ?
?
1
.
2 4 8
p p
= =
8. 
cos( ).cos( )
dx
I
x a x b
=
+ +
?
1 sin[( ) ( )]
sin( ) cos( ).cos( )
x a x b
a b x a x b
+ - +
=
- + +
?
1 sin( ).cos( ) cos( ).sin( )
sin( ) cos( ).cos( )
x a x b x a x b
a b x a x b
? ? + + - + +
=
? ?
- + +
? ?
?
[ ]
1
tan( ) tan( )
sin( )
x a x b dx
a b
= + - +
- ?
[ ]
1
logsec( ) log sec( )
sin( )
1 sec( )
log
sin( ) sec( )
x a x b c
a b
x a
c
a b x b
= + - + +
- ? ? +
= +
? ?
- +
? ?
Page 4


CBSE TEST PAPER-03 
CLASS - XII MATHEMATICS (Calculus: Integrals) 
Topic: - Integrals (ANSWERS) 
1. Put 
10
10
x
x t + =
9
(10 10 .log 10 )
x
e
x dx dt
dt
t
+ =
=
?
10
log | |
log( 10 )
x
t c
x c
= +
= + +
2. 
2 2 2
2 2
cos 2 2sin 1 2sin 2sin
 = 
cos cos
x x x x
dx dx
x x
+ - +
=
? ?
2
2
1
sec 
cos
tan
dx x dx
x
x c
= =
= +
? ?
3. Let 
2
( ) sin f x x =
2
2
( ) sin ( )
sin
f x x
x
- = - =
( ) f x =
? function is even 
4 4
2 2
4 0
sin 2 sin 
1 cos 2
2
2
x dx x dx
x
dx
p p
p - ? =
- ? ?
=
? ?
? ?
? ?
?
4
sin 2
2
1
4 2
o
x
x
p
p
? ?
= - ? ?
? ?
= - 4. 
5 4
log
3 2
 
x x
dx e
x x
?
?
? ? - ? =
? ?
?
- ? ?
?
? 
( )
4
1 x x - =
( )
2
1 x x - dx
?
2
x dx =
?
3
3
x
c = +
5. 
x
e
=
( )
x x
x
e e
e
- - ( )
x x
dx
e e
- +
?
( )
 
x x
x x
put e e t
e e dx dt
- - + =
- =
( )
log | |
log
x x
dt
t c
t
e e c
- = = +
= + +
?
6. 
3 3
6 6
cos
(1)
sin cos sin
1
cos
dx x
I dx
x x x
x
p p
p p
= = - - - - - - - +
+
? ?
 
3
6
cos
3 6
cos sin
3 6 3 6
x
I dx
x x
p
p
p p
p p p p
? ?
+ - ? ?
? ?
=
? ? ? ?
+ - + + - ? ? ? ?
? ? ? ?
?
( )
3
6
( )
sin
(2)
sin sin
b b
a a
f x dx f a b x dx
x
I dx
x x
p
p
?
= + - ?
?
= - - - - - - - +
? ?
?
?
3
6
(1) (2)
2 1 I dx
p
p
+
=
?
[ ]
3
6
3 6 6
x
p
p
p p p
=
= - =
 
12
I
p
=
7. 
4
4 4
sin .cos
cos sin
o
x x
I dx
x x
p
=
+
?
Dividing N and D by cos
4
x 
4 4
4 4
4 4
sin .cos
cos
cos sin
cos cos
o
x x
x
dx
x x
x x
p
=
+
?
2
4
4
2
4
2 2
tan .sec
1 tan
tan .sec
1 (tan )
o
o
x x
dx
x
x x
dx
x
p
p
=
+
=
+
?
?
2
2
 tan
2 tan .sec 
put x t
x x dx dt
=
=
1
2
1
1
1
2 1
1
tan
2
o
o
dt
t
t
- =
+
? ? =
? ?
?
1
.
2 4 8
p p
= =
8. 
cos( ).cos( )
dx
I
x a x b
=
+ +
?
1 sin[( ) ( )]
sin( ) cos( ).cos( )
x a x b
a b x a x b
+ - +
=
- + +
?
1 sin( ).cos( ) cos( ).sin( )
sin( ) cos( ).cos( )
x a x b x a x b
a b x a x b
? ? + + - + +
=
? ?
- + +
? ?
?
[ ]
1
tan( ) tan( )
sin( )
x a x b dx
a b
= + - +
- ?
[ ]
1
logsec( ) log sec( )
sin( )
1 sec( )
log
sin( ) sec( )
x a x b c
a b
x a
c
a b x b
= + - + +
- ? ? +
= +
? ?
- +
? ?
9. 
2
2
2 2
cos
cos 4sin
o
x
I dx
x x
p
=
+
?
( )
2
2
2 2
2
2
2 2
cos
cos 4 1 cos
cos
cos 4 4cos
o
o
x
dx
x x
x
dx
x x
p
p
=
+ - =
+ - ?
?
2
2
2
2
2
2
cos
4 3cos
1 3cos
3 4 3cos
o
o
x
dx
x
x
dx
x
p
p
=
- - - =
- ?
?
2
2
2
2
2
1 4 3cos 4
3 4 3cos
1 4
1
3 4 3cos
o
o
x
dx
x
dx
x
p
p
- - - =
- - ? ?
= - ? ?
- ? ?
?
?
[ ]
2 2
2
2
2
2
2
2 2
1 4
1
3 3 4 3cos
1 4 cos
4 3cos 3 3
cos cos
o o
o
o
dx
dx
x
dx x
x
x
x x
p p
p
p
- = +
- - = +
- ? ?
?
2
2
2
2
2
2
1 4 sec 
3 2 3 4sec 3
4 sec 
6 3 4(1 tan ) 3
o
o
x dx
o
x
x
dx
x
p
p
p
p
- ? ?
= - +
? ?
- ? ?
- = +
+ - ?
?
2
 tan
sec 
put x t
x dx dt
=
=
2
0
6 4(1 ) 3
1
6 4
dt
t
p
p
8
- = +
+ - - = +
?
4
.
0
2
1
3
4
dt
t
8
+
?
( )
1
1 1
1 1
. tan 2
6 3 2
2
tan tan
6 3
o
t
o
p
p
8
- - - - ? ? = +
? ?
- = + 8 -
Page 5


CBSE TEST PAPER-03 
CLASS - XII MATHEMATICS (Calculus: Integrals) 
Topic: - Integrals (ANSWERS) 
1. Put 
10
10
x
x t + =
9
(10 10 .log 10 )
x
e
x dx dt
dt
t
+ =
=
?
10
log | |
log( 10 )
x
t c
x c
= +
= + +
2. 
2 2 2
2 2
cos 2 2sin 1 2sin 2sin
 = 
cos cos
x x x x
dx dx
x x
+ - +
=
? ?
2
2
1
sec 
cos
tan
dx x dx
x
x c
= =
= +
? ?
3. Let 
2
( ) sin f x x =
2
2
( ) sin ( )
sin
f x x
x
- = - =
( ) f x =
? function is even 
4 4
2 2
4 0
sin 2 sin 
1 cos 2
2
2
x dx x dx
x
dx
p p
p - ? =
- ? ?
=
? ?
? ?
? ?
?
4
sin 2
2
1
4 2
o
x
x
p
p
? ?
= - ? ?
? ?
= - 4. 
5 4
log
3 2
 
x x
dx e
x x
?
?
? ? - ? =
? ?
?
- ? ?
?
? 
( )
4
1 x x - =
( )
2
1 x x - dx
?
2
x dx =
?
3
3
x
c = +
5. 
x
e
=
( )
x x
x
e e
e
- - ( )
x x
dx
e e
- +
?
( )
 
x x
x x
put e e t
e e dx dt
- - + =
- =
( )
log | |
log
x x
dt
t c
t
e e c
- = = +
= + +
?
6. 
3 3
6 6
cos
(1)
sin cos sin
1
cos
dx x
I dx
x x x
x
p p
p p
= = - - - - - - - +
+
? ?
 
3
6
cos
3 6
cos sin
3 6 3 6
x
I dx
x x
p
p
p p
p p p p
? ?
+ - ? ?
? ?
=
? ? ? ?
+ - + + - ? ? ? ?
? ? ? ?
?
( )
3
6
( )
sin
(2)
sin sin
b b
a a
f x dx f a b x dx
x
I dx
x x
p
p
?
= + - ?
?
= - - - - - - - +
? ?
?
?
3
6
(1) (2)
2 1 I dx
p
p
+
=
?
[ ]
3
6
3 6 6
x
p
p
p p p
=
= - =
 
12
I
p
=
7. 
4
4 4
sin .cos
cos sin
o
x x
I dx
x x
p
=
+
?
Dividing N and D by cos
4
x 
4 4
4 4
4 4
sin .cos
cos
cos sin
cos cos
o
x x
x
dx
x x
x x
p
=
+
?
2
4
4
2
4
2 2
tan .sec
1 tan
tan .sec
1 (tan )
o
o
x x
dx
x
x x
dx
x
p
p
=
+
=
+
?
?
2
2
 tan
2 tan .sec 
put x t
x x dx dt
=
=
1
2
1
1
1
2 1
1
tan
2
o
o
dt
t
t
- =
+
? ? =
? ?
?
1
.
2 4 8
p p
= =
8. 
cos( ).cos( )
dx
I
x a x b
=
+ +
?
1 sin[( ) ( )]
sin( ) cos( ).cos( )
x a x b
a b x a x b
+ - +
=
- + +
?
1 sin( ).cos( ) cos( ).sin( )
sin( ) cos( ).cos( )
x a x b x a x b
a b x a x b
? ? + + - + +
=
? ?
- + +
? ?
?
[ ]
1
tan( ) tan( )
sin( )
x a x b dx
a b
= + - +
- ?
[ ]
1
logsec( ) log sec( )
sin( )
1 sec( )
log
sin( ) sec( )
x a x b c
a b
x a
c
a b x b
= + - + +
- ? ? +
= +
? ?
- +
? ?
9. 
2
2
2 2
cos
cos 4sin
o
x
I dx
x x
p
=
+
?
( )
2
2
2 2
2
2
2 2
cos
cos 4 1 cos
cos
cos 4 4cos
o
o
x
dx
x x
x
dx
x x
p
p
=
+ - =
+ - ?
?
2
2
2
2
2
2
cos
4 3cos
1 3cos
3 4 3cos
o
o
x
dx
x
x
dx
x
p
p
=
- - - =
- ?
?
2
2
2
2
2
1 4 3cos 4
3 4 3cos
1 4
1
3 4 3cos
o
o
x
dx
x
dx
x
p
p
- - - =
- - ? ?
= - ? ?
- ? ?
?
?
[ ]
2 2
2
2
2
2
2
2 2
1 4
1
3 3 4 3cos
1 4 cos
4 3cos 3 3
cos cos
o o
o
o
dx
dx
x
dx x
x
x
x x
p p
p
p
- = +
- - = +
- ? ?
?
2
2
2
2
2
2
1 4 sec 
3 2 3 4sec 3
4 sec 
6 3 4(1 tan ) 3
o
o
x dx
o
x
x
dx
x
p
p
p
p
- ? ?
= - +
? ?
- ? ?
- = +
+ - ?
?
2
 tan
sec 
put x t
x dx dt
=
=
2
0
6 4(1 ) 3
1
6 4
dt
t
p
p
8
- = +
+ - - = +
?
4
.
0
2
1
3
4
dt
t
8
+
?
( )
1
1 1
1 1
. tan 2
6 3 2
2
tan tan
6 3
o
t
o
p
p
8
- - - - ? ? = +
? ?
- = + 8 - 2
6 3 2
6 3
6
o
p p
p p
p
- ? ?
= + - ? ?
? ?
- = +
=
10. 
2
logsin (1)
o
I x dx
p
= - - - - - - - - - ?
[
2
2
logsin          p4
2
log cos (2)
o
o
I x dx by
I x dx
p
p
p ? ?
= - ? ?
? ?
= - - - - - - - - - ?
?
( )
2
(1) (2)
2 logsin logcos
o
I x x dx
p
+
= +
?
( )
2
2 2
2
logsin .cos log 2 log 2
logsin log2
o
o o
x x dx
x dx dx
p
p p
= + - = - ?
? ?
 2
2
put x t
dt
dx
=
=
1
2 logsin log 2
2 2
2
o
I t dt
p
p
= - =
?
2
[
2
6
logsin log 2 
2
o
t dt byP
p
p
- ?
[
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logsin log 2         
2
2 log 2
2
log 2
2
o
o
x dx byP
I I
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p
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p
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= - = - = - ?
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