If u and v are any two differentiable functions of a single variable x (say) Then, by
the product rule of differentiation, we have
(
)
d
uv
dx
 =
dv
du
u
v
dx
dx
+
Integrating both sides, we get
uv =
dv
du
u
dx
v
dx
dx
dx
+
∫
∫
or
udv
dx
∫dx
 =
uv – vdu
dx
∫dx (1)
Let
u = f (x) and dv
dx = g(x) Then
du
dx = f ′(x) and v = 
( )
∫g x dx
324
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