## Summary of Trigonometric Substitution

### Essential Concepts

• For integrals involving $\sqrt{{a}^{2}-{x}^{2}}$, use the substitution $x=a\sin\theta$ and $dx=a\cos\theta d\theta$.
• For integrals involving $\sqrt{{a}^{2}+{x}^{2}}$, use the substitution $x=a\tan\theta$ and $dx=a{\sec}^{2}\theta d\theta$.
• For integrals involving $\sqrt{{x}^{2}-{a}^{2}}$, substitute $x=a\sec\theta$ and $dx=a\sec\theta \tan\theta d\theta$.

Glossary

trigonometric substitution
an integration technique that converts an algebraic integral containing expressions of the form $\sqrt{{a}^{2}-{x}^{2}}$, $\sqrt{{a}^{2}+{x}^{2}}$, or $\sqrt{{x}^{2}-{a}^{2}}$ into a trigonometric integral