Then dt = – 2x dx
326
MATHEMATICS
Therefore,
2
1
x dx
−x
∫
 =
1
2
dt
–
∫t
 = 
2
–
1
t
x
= −
−
Hence,
1
2
sin
1
–
x
x dx
−x
∫
 =
(
)
1
2
2
12
(sin
)
1
(
1
)
1
– x
–
x
–
x
dx
x
−
−
−
−
∫
=
2
1
1
sin
C
–
x
x
x
−
−
+
+
 = 
2
1
1
sin
C
x –
x
x
−
−
+
Alternatively, this integral can also be worked out by making substitution sin–1 x  = θ and
then integrating by parts Example 21  Find 
exsin
x dx
∫
Solution  Take ex as the first function and sin x as second function Then, integrating
by parts, we have
I
sin
(
cos )
cos
x
x
x
e
x dx
e
–
x
e
x dx
=
=
+
∫
∫
= – e x cos x + I1 (say) (1)
Taking ex
 and cos x as the first and second functions, respectively, in I1, we get
I1 =
sin
sin
x
x
e
x – e
x dx
∫
Substituting the value of I1 in (1), we get
I = – ex cos x + ex sin x – I  or  2I = ex (sin x – cos x)
Hence,
I =
sin
(sin
cos ) + C
2
x
x
e
e
x dx
x –
x
=
∫
Alternatively, above integral can also be determined by taking sin x as the first function
and ex the second function