Chapter 8: Techniques of Integration - Section 8.2 - Integration by Parts - Exercises 8.2 - Page 455: 14

$$\int 4x \sec^22x \ dx =2 x \tan 2 x + \ln |\cos 2x| +C $$

Given $$\int 4x \sec^22x \ dx $$ Using integration by parts, we get: $$ u=4 x \Rightarrow du=4dx $$ $$ dv=\sec^22x dx \Rightarrow v= \frac{1}{2}\tan 2x $$ So, we get \begin{aligned} I&=\int 4x \sec^22x \ dx\\ &=uv- \int vdu\\ &=2 x \tan 2 x - \int 2 \tan2x \ d x\\ &=2 x \tan 2 x - \int \frac{2\sin 2x}{\cos 2x}\ d x\\ &=2 x \tan 2 x + \ln |\cos 2x| +C \end{aligned}