Integration by Trigonometric Substitution

A process to integrate when you have one of the following expressions in the integrand:

• $\sqrt{a^2-x^2}$
• $\sqrt{a^2+x^2}$
• $\sqrt{x^2-a^2}$ Where: $a\in \mathbb{R}$. You will be able to Construct a Trig Substitution Triangle to relate each side to the equation.

You are able to use the Pythagorean Identities to get the following implications:

• $\sqrt{a^2-x^2}$. Choosing $x=a\sin \theta$ where $\theta \in [-\frac{\pi }{2},\frac{\pi }{2}]$ give us $\sqrt{a^2-x^2}=a\cos \theta$
• $\sqrt{a^2+x^2}$. Choosing $x=a\tan \theta$ where $\theta \in (-\frac{\pi }{2},\frac{\pi }{2})$ gives us $\sqrt{a^2+x^2}=a\sec \theta$
• $\sqrt{x^2-a^2}$. Choosing $x=a\sec \theta$ where $\theta \in [0,\frac{\pi }{2})\cup (\frac{\pi }{2},\pi ]$ gives us $\sqrt{x^2-a^2}=a\tan \theta$ Note at the end of the integration, you will get x in terms of $\theta$, so use Triangle Method to find the final value of $x$

Process

1. Use a triangle to make a substitution for x
2. Solve with Integration by Trigonometric Identities
3. Use the triangle to reverse the substitution

Examples

Example 1