5.5 | Integration by Substitution

Integration by Substitution: Evaluating $$\int f(g(x))g'(x) \; dx$$

1. Let $$u = g(x)$$, where $$g(x)$$ is part of the integrand, usually the “inside function” of the composite function.
2. Compare $$du = g'(x) \; dx$$.
3. Use the substitution $$u = g(x)$$ and $$du = g'(x) \; dx$$ to transform the integral into one that involves only $$u: \; \; \int f(u) \; du$$.
4. Find the resulting integral.
5. Replace $$u$$ by $$g(x)$$ so that the final solution is in terms of $$x$$.
\begin{aligned} & \int \tan u \; du = \ln | \sec u | + C \\ & \int \cot u \; du = \ln | \sin u | + C \\ & \int \sec u \; du = \ln | \sec u + \tan u| + C \\ & \int \csc u \; du = \ln | \csc u – \cot u| + C \\ \end{aligned}
\begin{aligned} & \int \frac{1}{\sqrt{1-u^2}} \; du = \sin^{-1} u + C \\ & \int \frac{1}{1+u^2} \; du = \tan^{-1} u + C \\ & \int \frac{1}{|u|\sqrt{u^2-1}} \; du = \sec^{-1} |u| + C \\ \end{aligned}