Q. 60 - Calculus

Question

Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)

$\int \frac{{\sec }^{2}x}{\tan x+1}dx$

Step-by-Step Solution

Answer

The solution of the given integral is $\int \frac{{\sec }^{2}x}{\tan x+1}dx=\ln \left(\tan x+1\right)+C$.

Step 1. Given Information

Solving the given integrals.

$\int \frac{{\sec }^{2}x}{\tan x+1}dx$

Step 2. Using the substitution method.

Let

$$\begin{gathered} u=\tan x+1 \\ \frac{du}{dx}={\sec }^{2}x \\ du={\sec }^{2}xdx \end{gathered}$$

Step 3. This substitution changes the integral into

$$\begin{gathered} \int \frac{{\sec }^{2}x}{\tan x+1}dx=\int \frac{1}{u}du \\ \int \frac{{\sec }^{2}x}{\tan x+1}dx=\ln \left(u\right)+C \\ \int \frac{{\sec }^{2}x}{\tan x+1}dx=\ln \left(\tan x+1\right)+C \end{gathered}$$