Summary of Trigonometric Substitution

Essential Concepts

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

Glossary

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