Integration/Substitution/Remark

The substitution is applied in the following way: suppose that the integral

$\int _{a}^{b}f(t)\,dt$

has to be computed. Then one needs an idea that the integral gets simpler by the substitution

${}t=\varphi (s)\,$

(taking into account the derivative ${}\varphi '(s)$ and that the inverse function ${}\varphi ^{-1}$ has to be determined). Setting ${}c=\varphi ^{-1}(a)$ and ${}d=\varphi ^{-1}(b)$, we have the situation

$[c,d]{\stackrel {\varphi }{\longrightarrow }}[a,b]{\stackrel {f}{\longrightarrow }}\mathbb {R} .$

In certain cases, some standard substitutions help.

In order to make a substitution, three operations have to be done.

1. Replace ${}f(t)$ by ${}f(\varphi (s))$.
2. Replace ${}dt$ by ${}\varphi '(s)ds$.
3. Replace the integration bounds ${}a$ and ${}b$ by ${}\varphi ^{-1}(a)$ and ${}\varphi ^{-1}(b)$.

To remember the second step, think of

${}dt=d\varphi (s)=\varphi '(s)ds\,,$

which in the framework "differential forms“, has a meaning.