Question

Solve each of the integrals in Exercises 39–74. Some integrals require trigonometric substitution, and some do not. Write your answers as algebraic functions whenever possible.

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

Short answer

The solution of the integral is$\frac{1}{3}{\left(x^2+1\right)}^{\frac{3}{2}}+C.$

Step-by-step solution

Step 1. Given Information.

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

Step 2. Solve. 

To solve the integral, let $u=x^2+1,$ so derivation of uis $du=2xdx.$

Thus, substitute u into the original integral,

$$\begin{gathered} \int x\sqrt{x^2+1}dx \\ =\int \frac{1}{2}\sqrt{u-1}\sqrt{u}\frac{du}{\sqrt{u-1}} \\ =\frac{1}{2}\int \sqrt{u}du \\ =\frac{1}{2}\left[\frac{2}{3}{u}^{\frac{3}{2}}\right]+C \\ =\frac{1}{3}{u}^{\frac{3}{2}}+C \end{gathered}$$

Substitute back uin the above equation,

$=\frac{1}{3}{\left(x^2+1\right)}^{\frac{3}{2}}+C$