Summary of Trigonometric Substitution

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