Tutorial 4: Trigonometric Substitution

Tutorial problems focused on trigonometric substitution for integrals containing radicals.

Sections

Basic Substitutions (Problems 1-4)

  • $\sqrt{a^2 - x^2}$: $x = a\sin\theta$
  • $\sqrt{a^2 + x^2}$: $x = a\tan\theta$
  • $\sqrt{x^2 - a^2}$: $x = a\sec\theta$

More Complex Integrals (Problems 5-8)

  • Completing the square first
  • Multiple substitutions
  • Integrals requiring back-substitution

Definite Integrals (Problems 9-12)

  • Changing limits with substitution
  • Evaluating without back-substitution

Substitution Summary

Radical Substitution Identity Triangle
$\sqrt{a^2 - x^2}$ $x = a\sin\theta$ $\cos^2\theta = 1 - \sin^2\theta$ sin = opp/hyp
$\sqrt{a^2 + x^2}$ $x = a\tan\theta$ $\sec^2\theta = 1 + \tan^2\theta$ tan = opp/adj
$\sqrt{x^2 - a^2}$ $x = a\sec\theta$ $\tan^2\theta = \sec^2\theta - 1$ sec = hyp/adj

Common Results

  • $\int \frac{dx}{\sqrt{a^2 - x^2}} = \arcsin\frac{x}{a} + C$
  • $\int \frac{dx}{a^2 + x^2} = \frac{1}{a}\arctan\frac{x}{a} + C$

Links