(1)
Let
u = f (x) and dv
dx = g(x) Then
du
dx = f ′(x) and v = 
( )
∫g x dx
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MATHEMATICS
Therefore, expression (1) can be rewritten as
( ) ( )
∫f x g x dx
 =
( )
( )
[
( )
]
( )
f x
g x dx –
g x dx f
′x dx
∫
∫ ∫
i e ,
( ) ( )
∫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)