Chapter 8 - Techniques of Integration - 8.2 Trigonometric Integrals - Exercises - Page 403: 48

$$\tan e^x-e^x +c$$

Given $$\int e^x\tan^2(e^x)dx $$ Let $$ u =e^x\ \ \ \to \ \ \ du=e^xdx $$ Then \begin{align*} \int e^x\tan^2(e^x)dx &= \int \tan^2udu\\ &= \int (\sec^2u-1)du\\ &=\tan u-u +c\\ &= \tan e^x-e^x +c \end{align*}