Integration by Substitution
The method of substitution can often be used to simplify
complicated integration problems. It is based on the chain rule
for differentiation.
Suppose that
F
is an antiderivative of
f
, and
g
is a
differentiable function. Then
by the chain rule.
In integral form, we can write this as
( ( ))
( ( ))
( )
d
F g x
F g x g x
dx
′
′
=
( ( ))
( )
( ( ))
F g x g x dx
F g x
C
′
′
=
+
∫
( ( ))
( )
( ( ))
(1)
f g x g x dx
F g x
C
′
=
+
∫
or