Question

Use integration formulas to solve each integral in Exercises 21–62. You may have to use algebra, educated guess-and-check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating.

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

Short answer

The solution of the integral is$\ln \left|1+x^2\right|+C.$

Step-by-step solution

Step 1. Given Information.  

The given integral is$\int \frac{2x}{1+x^2}dx.$

Step 2. Solve. 

By solving the integral we get,

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

Step 3. Verification. 

To verify the answer we differentiate $\ln \left|1+x^2\right|+C$it.

On differentiating we get,

$$\begin{gathered} \ln \left|1+x^2\right|+C \\ =\frac{d}{dx}\ln \left|1+x^2\right|+\frac{d}{dx}C \\ =\frac{1}{1+x^2}\frac{d}{dx}{x}^2+0 \\ =\frac{2x}{1+x^2} \end{gathered}$$

Hence proved.