Q. 59 - 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.)

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

Step-by-Step Solution

Answer

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

Step 1. Given Information

Solving the given integrals.

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

Step 2. Using the substitution method.

Let

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

Step 3. This substitution changes the integral into

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