Question

Integrate the function $x{\sec }^{2}x$

Answer 1

Let $I=\int x{\sec }^{2}xdx$
Taking $x$ as first function and ${\sec }^{2}x$ as second function and integrating by parts, we obtain
$I=x\int {\sec }^{2}xdx-\int \{\left\{\frac{d}{dx}x\right\}\int {\sec }^{2}xdx\}dx$
$=x\tan x-\int 1\cdot \tan xdx$
$=x\tan x+\log |\cos x|+C$