Irrational integrals

Integrals containing roots are almost always tackled with the 5-step substitution, choosing $t$ according to the type of root present:

• $\sqrt{ax+b}$: set $t = ax+b$.
• $\sqrt{f(x)}$ with $f'(x)$ appearing in the integrand: set $t = f(x)$.
• $\sqrt{a^2-x^2}$: use the trigonometric substitution $x = a\sin\theta$.

The criterion is always the same: choose the substitution that makes the root disappear, reducing the integral to an elementary form.

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Topics: Integrale
Concepts: Integrale per sostituzione
Methods: Integrale sostituzione
Skills: Integrare · Ragionare per casi