Question

Evaluate the Integral: \(\int {({{\tan }^2}} x + {\tan ^4}x)dx\)

Short answer

To evaluate the Integral\(\int {({{\tan }^2}} x + {\tan ^4}x)dx\)we will use identity\(1 + {\tan ^2}x = {\sec ^2}x\)thenthe substitution method to integrate it.

\(\int {({{\tan }^2}} x + {\tan ^4}x)dx = \frac{1}{3}{\tan ^3}x + c\)

Step-by-step solution

Taking \({\tan ^2}x\) common

Let \(\begin{aligned}{l}I &= \int {({{\tan }^2}} x + {\tan ^4}x)dx\\ &= \int {{{\tan }^2}} x(1 + {\tan ^2}x)dx\end{aligned}\)

Using Trigonometric Identity

As\(1 + {\tan ^2}x = {\sec ^2}x\)

\(I = \int {{{\tan }^2}} x{\sec ^2}xdx\)

Substitution Method

Put\(\tan x = u( = ){\sec ^2}xdx = du\)

\(\begin{aligned}{l}I &= \int {{u^2}} du\\I &= \frac{{{u^3}}}{3} + c\end{aligned}\)

Re-Substitute the value of \(u = \tan x\)

\(I = \frac{{{{\tan }^3}x}}{3} + c\)

Hencethe value of Integral:

\(\int {({{\tan }^2}} x + {\tan ^4}x)dx = \frac{1}{3}{\tan ^3}x + c\)