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 \(x\).
$$
\begin{aligned}
& \int \tan u \; du = \ln | \sec u | + C \\
& \int \cot u \; du = \ln | \sin u | + C \\
& \int \sec u \; du = \ln | \sec u + \tan u| + C \\
& \int \csc u \; du = \ln | \csc u – \cot u| + C \\
\end{aligned}
$$
$$
\begin{aligned}
& \int \frac{1}{\sqrt{1-u^2}} \; du = \sin^{-1} u + C \\
& \int \frac{1}{1+u^2} \; du = \tan^{-1} u + C \\
& \int \frac{1}{|u|\sqrt{u^2-1}} \; du = \sec^{-1} |u| + C \\
\end{aligned}
$$

E 5.5 Exercises