Consider
I =
( )
f x dx
∫
Put x = g(t) so that dx
dt  = g′(t) We write
dx = g′(t) dt
Thus
I =
( )
( ( ))
( )
f x dx
f g t
g t dt
=
′
∫
∫
This change of variable formula is one of the important tools available to us in the
name of integration by substitution It is often important to guess what will be the useful
substitution Usually, we make a substitution for a function whose derivative also occurs
in the integrand as illustrated in the following examples