Lecture #9: Trigonometric Substitution

Overview of Trigonometric Substitution

Integrals involving \(\sqrt{a^2 - x^2}\), \(\sqrt{a^2 + x^2}\) and \(\sqrt{x^2 - a^2}\) can often be solved with a trigonometric substitution using the Pythagorean identity:

\[ \begin{aligned} a \cos^2(\theta) + a \sin(\theta) = a \end{aligned} \]

These substitutions are:

\[ \begin{aligned} \sqrt{a^2 - x^2} \quad \Longrightarrow \quad x = a \sin(\theta) \quad \Longrightarrow \quad dx = \,???\\ \sqrt{a^2 + x^2} \quad \Longrightarrow \quad x = a \tan(\theta) \quad \Longrightarrow \quad dx = \,??? \\ \sqrt{x^2 - a^2} \quad \Longrightarrow \quad x = a \sec(\theta) \quad \Longrightarrow \quad dx = \,??? \end{aligned} \]

The Bad News

This is a great trick for solving integrals that are very tricky to solve, or even completely unsolvable otherwise.

However, the end result is a trigonometric integral in \(\theta\).

After you solve the trigonometric integral in \(\theta\), you have to convert it back to a solution in terms of \(x\).

Don’t forget your inverse trig functions!

Examples: Trig Substitution

Example: Evaluate \(\displaystyle\int \dfrac{1}{\sqrt{4 - x^2}} \, dx\).

Examples: Trig Substitution

Example: Evaluate \(\displaystyle\int \dfrac{1}{\sqrt{4 + x^2}} \, dx\).

Examples: Trig Substitution

Example: Evaluate \(\displaystyle\int \dfrac{1}{\sqrt{x^2 + 4x - 12}} \, dx\).