Chapter 7.2, Problem 34E

To determine

To evaluate the integral .

Answer to Problem 34E

The value of the ${\displaystyle \int {\tan }^7\left(\frac{x}{2}\right)}{\text{ sec}}^2\left(\frac{x}{2}\right)dx=\frac{{\tan }^8\left(\frac{x}{2}\right)}{4}+C$

Explanation of Solution

Given information:

The integral is ${\displaystyle \int {\tan }^7\left(\frac{x}{2}\right)}{\text{ sec}}^2\left(\frac{x}{2}\right)dx$

Calculation:

The integral is ${\displaystyle \int {\tan }^7\left(\frac{x}{2}\right)}{\text{ sec}}^2\left(\frac{x}{2}\right)dx$

Using substitution method of integration

  $\begin{array}{l}{\displaystyle \int {\tan }^7\left(\frac{x}{2}\right)}{\text{ sec}}^2\left(\frac{x}{2}\right)dx\\ \text{let tan}\left(\frac{x}{2}\right)=t\\ \text{differentiating both side}\\ \frac{1}{2}{\text{sec}}^2\left(\frac{x}{2}\right)dx=dt\\ dx=\frac{2}{{\text{sec}}^2\left(\frac{x}{2}\right)}dt\\ {\displaystyle \int t^7}{\text{ sec}}^2\left(\frac{x}{2}\right)\times \frac{2}{{\text{sec}}^2\left(\frac{x}{2}\right)}dt\\ {\displaystyle \int 2t^7}dt=\frac{2t^8}{8}+C=\frac{{\tan }^8\left(\frac{x}{2}\right)}{4}+C,\text{where }C\text{ is constant of integration}\text{.}\end{array}$

Hence ,the value of the ${\displaystyle \int {\tan }^7\left(\frac{x}{2}\right)}{\text{ sec}}^2\left(\frac{x}{2}\right)dx=\frac{{\tan }^8\left(\frac{x}{2}\right)}{4}+C$ .