Trigonometric Substitution

The central idea

Trigonometric substitution uses identities to simplify quadratic radicals: for $\sqrt{a^2-x^2}$ set $x=a\sin\theta$; for $\sqrt{a^2+x^2}$ set $x=a\tan\theta$; for $\sqrt{x^2-a^2}$ set $x=a\sec\theta$. Choose an angle interval so signs and inverse substitutions are controlled.

Definitions, hypotheses, and notation

Hyperbolic substitutions can sometimes avoid absolute-value complications, but the trig forms align with standard introductory identities. Completing the square first extends the method to quadratics such as $x^2+4x+13=(x+2)^2+9$. The shifted variable, not the original $x$, then matches the substitution.

For definite integrals, transform bounds to angles and remain in $\theta$, or back-substitute before applying $x$-bounds. As in ordinary substitution, mixing coordinate systems in the bounds is invalid.

Conceptual meaning

Each substitution converts a sum or difference of squares into a perfect trig square using $1-\sin^2=\cos^2$, $1+\tan^2=\sec^2$, or $\sec^2-1=\tan^2$. A reference triangle translates the final trig expression back to $x$.

A dependable method and decision rule

1. Complete the square or factor constants to match a standard radical.
2. Choose the corresponding trig substitution and compute $dx$.
3. Simplify the radical with an identity, respecting absolute values.
4. Integrate in $\theta$.
5. Back-substitute using an inverse trig function or labelled triangle, then check the original domain.

Fully worked example

Common mistakes and why they fail