Example about \( {sec}^{4}x\)

Example:

Compute , $\int{{{\sec }^{4}}x\,dx}$

Solution: we know that $\int{{{\sec }^{4}}x\,dx}=\int{{{\sec }^{2}}x{{\sec }^{2}}x\,dx}$ but ${{\sec }^{2}}x=1+{{\tan }^{2}}x$

So $\int{{{\sec }^{4}}x\,dx}=\int{\left( 1+{{\tan }^{2}}x \right){{\sec }^{2}}x\,dx}$ take $u=\tan x\Rightarrow du={{\sec }^{2}}x\,dx$

Thus $\int{{{\sec }^{4}}x\,dx}=\int{\left( 1+{{u}^{2}} \right)du}=u+\frac{1}{3}{{u}^{3}}+c=\tan x+\frac{1}{3}{{\tan }^{3}}x+c$