Chapter 5.7, Problem 10E

To determine

To calculate: The value of given integral by the use of the substitution $u=\tan x$

Answer to Problem 10E

The value of the given integral is $\frac{1}{9}{\tan }^{9}x+\frac{2}{7}{\tan }^{7}x+\frac{1}{5}{\tan }^{5}x+C$

Explanation of Solution

Given information:

The integral is ${\displaystyle \int {\tan }^{4}x{\sec }^{6}xdx}$

Formula used:

  ${\sec }^{2}x=1+{\tan }^{2}x$

  ${\displaystyle \int x^{n}dx=\frac{x^{n+1}}{n+1}}+C$

Calculation:

Use the substitution rule,

  $\begin{array}{l}u=\tan x\\ du={\sec }^{2}xdx\end{array}$

Now, the value of the given integral can be obtained as,

  $\begin{array}{l}{\displaystyle \int {\tan }^{4}x{\sec }^{6}xdx}\left[\text{original equation}\right]\\ ={\displaystyle \int {\tan }^{4}x{\left(1+{\tan }^{2}x\right)}^2{\sec }^{2}xdx}\left[{\sec }^{2}x=1+{\tan }^{2}x\right]\\ ={\displaystyle \int u^4{\left(1+u^2\right)}^{2}du\left[\text{substitute '}u\text{'}\text{ into the integral}\right]}\end{array}$

Rewrite the integral as shown in order to simplify integration

  $\begin{array}{l}{\displaystyle \int u^4\left(1+2u^2+u^4\right)du}\\ {\displaystyle \int \left(u^8+2u^6+u^4\right)du}\end{array}$

Integrate, then substitute $u=\tan x$ back into the equation to get the final answer

  $\begin{array}{l}{\displaystyle \int \left(u^8+2u^6+u^4\right)du}\\ =\frac{u^9}{9}+\frac{2u^7}{7}+\frac{u^5}{5}+C\\ =\frac{1}{9}{\tan }^{9}x+\frac{2}{7}{\tan }^{7}x+\frac{1}{5}{\tan }^{5}x+C\end{array}$

Therefore, the value of the given integral is $\frac{1}{9}{\tan }^{9}x+\frac{2}{7}{\tan }^{7}x+\frac{1}{5}{\tan }^{5}x+C$