Inverse function integration is an indefinite integration technique using an inverse function.
If 
is a strictly monotone real
function with a continuous derivative on a closed interval, and 
is its inverse function,
then integration by parts gives
 |
(1)
|
so
 |
(2)
|
where
 |
(3)
|
The identity 
also holds for any continuous
function 
that is strictly monotone
on an interval. Neither 
nor its inverse function
needs to be differentiable (Bensimhoun 2013, pp. 2-4).
Therefore, if it is possible to find an inverse 
of 
, integrate 
, make the replacement 
, and subtract the result from 
to obtain the result for the original integral 
.
If 
and 
are elementary on some closed interval, then 
is elementary iff 
is elementary.
