,
( ) ( )
∫f x g x dx
 =
( )
( )
[
( )
( )
]
f x
g x dx –
f
x
g x dx dx
′
∫
∫
∫
If we take f as the first function and g as the second function, then this formula
may be stated as follows:
“The integral of the product of two functions = (first function) × (integral
of the second function) – Integral of [(differential coefficient of the first function)
× (integral of the second function)]”
Example 17 Find 
xcos
x dx
∫
Solution Put f (x) = x (first function) and g (x) = cos x (second function) Then, integration by parts gives
xcos
x dx
∫
 =
cos
[
( ) cos
]
d
x
x dx –
x
x dx dx
dx
∫
∫
∫
=
sin
sin
x
x –
x dx
∫
 = x sin x + cos x + C
Suppose, we take
f (x) = cos x and g(x) = x 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