Question

Solve the following integral.

$\int se{c}^{6}xdx$

Short answer

Answer is$\frac{{\tan }^{5}x}{5}+\frac{2{\tan }^{3}x}{3}+\tan x+C$

Step-by-step solution

Step 1. Given information

An integral is$\int se{c}^{6}xdx$

Step 2. Explanation

$\begin{array}{rcl}\int se{c}^{6}xdx& =& \int se{c}^{2}xse{c}^{4}xdx\\ & =& \int {\left({\tan }^{2}x+1\right)}^{2}se{c}^{2}xdx\end{array}$

Let $\tan x=u$

$\begin{array}{rcl}\int se{c}^{6}xdx& =& \int {\left(u^2+1\right)}^{2}du\\ & =& \int u^4+2u^2+1du\\ & =& \frac{u^5}{5}+\frac{2u^3}{3}+u+C\\ & =& \frac{{\tan }^{5}x}{5}+\frac{2{\tan }^{3}x}{3}+\tan x+C\end{array}$