Chapter 7.3, Problem 58E

To determine

To determine the value of ${\displaystyle \int co{\sec }^3(x)dx}$

Answer to Problem 58E

The value of the integral ${\displaystyle \int co{\sec }^3(x)dx}$ is $co{\sec }^2(x).\left(-\ln (co\sec (x)+\cot (x))\right)-2co{\sec }^3(x)\cot (x)+C$

Explanation of Solution

Given information: ${\displaystyle \int co{\sec }^3(x)dx}$

Formula used: ${\displaystyle \int co\sec (x)dx}=-\ln (co\sec (x)+\cot (x))+C$

Calculation :

  ${\displaystyle \int co{\sec }^3(x)dx}={\displaystyle \int co\sec (x)co{\sec }^2(x)dx}$

Using integration by parts, the expression becomes as:

  $\begin{array}{l}{\displaystyle \int co\sec (x)co{\sec }^2(x)dx}=co{\sec }^2(x).\left(-\ln (co\sec (x)+\cot (x))\right)+\\ 2co\sec (x)\cos ec(x)\cot (x).\left(\frac{-\cos ec(x)\cot (x)-\cos e{c}^2(x)}{\cos ec(x)+\cot (x)}\right)+C\end{array}$

  $=co{\sec }^2(x).\left(-\ln (co\sec (x)+\cot (x))\right)-2co{\sec }^3(x)\cot (x)+C$

Hence, ${\displaystyle \int co{\sec }^3(x)dx}=co{\sec }^2(x).\left(-\ln (co\sec (x)+\cot (x))\right)-2co{\sec }^3(x)\cot (x)+C$