Chapter 8 - Techniques of Integration - 8.2 Trigonometric Integrals - Exercises - Page 404: 71

$$\int \tan ^{k} x d x =\frac{\tan^{k-1}}{k-1}- \int \tan ^{k-2} x d x $$

\begin{aligned} \int \tan ^{k} x d x &=\int \tan ^{k-2} x \tan ^{2} x d x \\ &=\int \tan ^{k-2} x\left(\sec ^{2} x-1\right) d x \\ &=\int \tan ^{k-2} x \sec ^{2} x d x-\int \tan ^{k-2} x d x \\ &=\frac{\tan^{k-1}}{k-1}- \int \tan ^{k-2} x d x \end{aligned}