Question

Evaluate: $\int x{\sec }^{2}xdx$

Answer 1

$I=\int x{\sec }^{2}xdx$

We know that

$\int uv=u\int v-\int (\int v)\frac{du}{dx}.dx$

$u=x$ $v={\sec }^{2}x$

$I=x\int {\sec }^{2}x-\int (\int {\sec }^{2}x).1dx$

$I=x\tan x-\int \tan xdx$

$I=x\tan x-\ln |\sec x|+c$.