Trigonometric substitution

Summary

Trigonometric substitution is a substitution method for integrals that contain $\sqrt{a^2 - x^2}$, $\sqrt{a^2 + x^2}$, or $\sqrt{x^2 - a^2}$. We set $x$ equal to $a \sin \theta$, $a \tan \theta$, or $a \sec \theta$. The Pythagorean identity then removes the square root.

Remarks

With $x = a \sin \theta$ for $\theta \in [-\pi/2, \pi/2]$, we get $\sqrt{a^2 - x^2} = a \cos \theta$, for $a > 0$. For example, this substitution gives $\int_{-1}^{1} \sqrt{1 - x^2} \, dx = \pi/2$, the area of a half disc. Hyperbolic functions can replace the trigonometric functions in the second and the third case.

Explanations

L1 human

A square root such as $\sqrt{1 - x^2}$ is the side of a right triangle. Writing $x$ as the sine of an angle turns the root into a cosine.