Trigonometric Substitution

One way that we can use the Change of Variables Theorem to transform an integral is to make the variable we are integrating a function of a new variable, and then convert it into an integral over this new variable.

\int g(x) dx = \int g(u(t)) \cdot u'(t) dt \quad \text{where } x = u(t)

Even though it looks like we are going from a simple integral to a more complicated one, we can sometimes use special properties of the functions we choose for our substitution to cancel out different terms and turn what seems like a complicated integral into a much simpler one. In particular, there are certain integrals we can make simple by choosing $u(t)$ to be a trigonometric or hyperbolic function.

This technique is called trigonometric substitution, and we will show how it can be used to solve, for example, the integral $\int \frac1{\sqrt{1-x^2}} dx$

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