Unit 7: Trigonometry Substitution for Indefinite Integrals

Trig substitution is a technique used in calculus to evaluate indefinite integrals that involve expressions with square roots, particularly those of the form [latex]\sqrt{a^2-b^2x^2}, \sqrt{a^2+b^2x^2},[/latex] [latex]\sqrt{b^2x^2-a^2}[/latex]. Trigonometric substitution involves replacing these expressions with trigonometric identities — such as [latex]x=\frac{a}{b}\sin\theta,[/latex] [latex]x=\frac{a}{b}\tan\theta, x=\frac{a}{b}\sec\theta[/latex] — we can simplify these expressions into forms that are easier to integrate.

The process of trigonometric substitution typically involves three main steps.

(1) Choose the appropriate trig substitution: Identify the form of the square root expression in the integral, and select the corresponding trig substitution (refer to the table below).

(2) Rewrite the integral in terms of [latex]\theta[/latex]: replace [latex]x[/latex] with the chosen trig substitution. Simplify the integral using trig identities to eliminate square roots, which often reduces the integral to a simpler trigonometric form that’s easier to evaluate.

(3) Integrate and Back-Substitute: Evaluate the integral in terms of [latex]\theta[/latex]. Once you have the antiderivative, use inverse trigonometric functions to convert [latex]\theta[/latex] back to the original variable [latex]x[/latex], yielding the final solution in terms of  [latex]x[/latex].

Practice Questions:

Evaluate the following indefinite integrals before referring to the video for comprehensive solutions.

Q1: [latex]\int \frac{1}{\sqrt{4+x^2}}dx[/latex]

Q2: [latex]\int \frac{x^2}{\sqrt{9-x^2}}dx[/latex]