Integration/Substitution/Remark

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

(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.

  1. Replace by .
  2. Replace by .
  3. Replace the integration bounds and by and .

To remember the second step, think of

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