Find the integral of x (1+x ²)^1/2

by RoRi

October 31, 2015

Evaluate the following integral:

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

We use the substitution

$u = 1+x^2 \qquad \implies \qquad du = 2x \, dx.$

Then we have

$\begin{align*} \int x \sqrt{1+x^2} \, dx &= \frac{1}{2} \int \sqrt{u} \, du \\ &= \frac{1}{2} \left(\frac{2}{3} u^{\frac{3}{2}}\right) + C \\ &= \frac{1}{3} (1+x^2)^{\frac{3}{2}} + C. \end{align*}$

Stumbling Robot

Find the integral of x (1+x ²)^1/2

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