Chapter 7.2, Problem 40E

To determine

To determine the integral ${\displaystyle \int {\tan }^{2}x{\sec }^{2}xdx}$ by substitution.

Answer to Problem 40E

The value of the integral ${\displaystyle \int {\tan }^{2}x{\sec }^{2}xdx}$ is $\frac{{(\tan x)}^3}{3}+C$

Explanation of Solution

Given information: ${\displaystyle \int {\tan }^{2}x{\sec }^{2}xdx}$

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

Calculation : Put $\tan x=t$ so that ${\sec }^{2}xdx=dt$

The given integral becomes as:

  ${\displaystyle \int t^{2}dt=\frac{t^3}{3}+C}$

Hence, ${\displaystyle \int {\tan }^{2}x{\sec }^{2}xdx}=\frac{{(\tan x)}^3}{3}+C$