Then
xcos
x dx
∫
 = cos
[
(cos )
]
d
x
x dx –
x
x dx dx
dx
∫
∫
∫
= (
)
2
2
cos
sin
2
2
x
x
x
x
dx
+∫
Thus, it shows that the integral 
xcos
x dx
∫
 is reduced to the comparatively more
complicated integral having more power of x Therefore, the proper choice of the first
function and the second function is significant Remarks
(i)
It is worth mentioning that integration by parts is not applicable to product of
functions in all cases For instance, the method does not work for 
xsin
x dx
∫