Irrational and trigonometric substitutions

Substitution is particularly useful with roots: one chooses $t$ equal to the expression under the root, or — when $\sqrt{a^2-x^2}$ appears — one switches to an angular variable.

Example — Irrational integral: $\int\dfrac{x}{\sqrt{2x^2+1}}\,dx$

Set $t = 2x^2+1$, $dt = 4x\,dx$: $\frac{1}{4}\int t^{-1/2}\,dt = \frac{1}{2}\sqrt{t} = \frac{1}{2}\sqrt{2x^2+1}+C.$

Example — Trigonometric substitution: $\int\sqrt{1-x^2}\,dx$

Set $x = \sin\theta$, $dx = \cos\theta\,d\theta$, $\sqrt{1-x^2} = \cos\theta$: $\int\cos^2\theta\,d\theta = \frac{\theta}{2}+\frac{\sin(2\theta)}{4}+C = \frac{\arcsin x}{2}+\frac{x\sqrt{1-x^2}}{2}+C.$

Trigonometric substitution exploits the identity $1-\sin^2\theta = \cos^2\theta$ to “eliminate” the root, transforming the irrational integral into a trigonometric one.

Links

Topics: Integrale
Concepts: Integrale per sostituzione
Methods: Integrale sostituzione
Skills: Calcolare · Integrare