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 7 6