The
substitution
is applied in the following way: suppose that the integral
-
has to be computed. Then one needs an idea that the integral gets simpler by the substitution
-
![{\displaystyle {}t=\varphi (s)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2c64c261a3a4687ffcd151e55b06b41b7ea55a8)
(taking into account the derivative
and that the inverse function
has to be determined).
Setting
and
,
we have the situation
-
In certain cases, some standard substitutions help.
In order to make a substitution, three operations have to be done.
- Replace
by
.
- Replace
by
.
- Replace the integration bounds
and
by
and
.
To remember the second step, think of
-
![{\displaystyle {}dt=d\varphi (s)=\varphi '(s)ds\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/259ae7475096e895117bd5611ef4bbd54b69a02b)
which in the framework "differential forms“, has a meaning.