20 The value of 
3
5
2
2
(
cos
tan
1)
x
x
x
x
dx
π
−π
+
+
+
∫
 is
(A) 0
(B) 2
(C) π
(D) 1
21 The value of 
02
4
3sin
log
4
3cos
x
dx
x
π


+


+


∫
 is
(A) 2
(B)
43
(C) 0
(D) –2
348
MATHEMATICS
Miscellaneous Examples
Example 37 Find cos 6
1 sin 6
x
x dx
+
∫
Solution Put t = 1 + sin 6x, so that dt = 6 cos 6x dx
Therefore
21
1
cos 6
1 sin 6
6
x
x dx
t dt
+
=
∫
∫
=
3
3
2
2
1
2
1
( )
C = 
(1 sin 6 )
C
6
3
9
t
x
×
+
+
+
Example 38 Find 
1
4
4
5
(
)
x
x
dx
−x
∫
Solution We have 
1
1
4
4
4
3
5
4
1
(1
)
(
)
x
x
x
dx
dx
x
x
−
−
=
∫
∫
Put 
– 3
3
4
1
3
1
1–
, so that
x
t
dx
dt
x
x
−
=
=
=
Therefore 
1
1
4
4
4
5
(
)
31
x
x
dx
t
dt
−x
=
∫
∫
 = 
5
5
4
4
3
1
4
4
1
C =
1
C
3
5
15
t
x


×
+
−
+




Example 39 Find 
4
2
(
1) (
1)
x dx
x
x
−
+
∫
Solution We have
4
2
(
1)(
1)
x
x
x
−
+
 =
3
12
(
1)
1
x
x
x
x
+
+
−
+
−
=
12
(
1)
(
1) (
1)
x
x
x
+
+
−
+ (1)
Now express
2
1
(
1)(
1)
x
x
−
+
 =
2
A
B
C
(
1)
(
1)
x
x
x
+
+
−
+