Q. 23 - 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.)

$\int \frac{8x}{x^2+1}\mathrm{d}x$

Step-by-Step Solution

Answer

The solution of the given integral is $\int \frac{8x}{x^2+1}\mathrm{d}x=4\log (x^2+1)+C$.

Step 1. Given Information

Solving the given integrals.
$\int \frac{8x}{x^2+1}\mathrm{d}x$

Step 2. Solving the given integral using substitution method.

Let

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

Step 3. This substitution changes the integral into

$$\begin{gathered} \int \frac{8x}{x^2+1}\mathrm{d}x=\frac{8}{2}\int \frac{1}{u}\mathrm{du} \\ \int \frac{8x}{x^2+1}\mathrm{d}x=4\log u+C \\ \int \frac{8x}{x^2+1}\mathrm{d}x=4\log (x^2+1)+C \end{gathered}$$