# How do you evaluate the integral int x/(root3(x^2-1))?

Apr 13, 2017

The integral equals $\frac{3}{4} {\left({x}^{2} - 1\right)}^{\frac{2}{3}} + C$

#### Explanation:

We will use u-substitution for this integral. Let $u = {x}^{2} - 1$.

Then $\mathrm{du} = 2 x \mathrm{dx}$ and $\mathrm{dx} = \frac{\mathrm{du}}{2 x}$. Call the integral $I$.

$I = \int \frac{x}{\sqrt[3]{u}} \cdot \frac{\mathrm{du}}{2 x}$

$I = \frac{1}{2} \int \frac{1}{\sqrt[3]{u}}$

$I = \frac{1}{2} \int {u}^{- \frac{1}{3}}$

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

$I = \frac{3}{4} {u}^{\frac{2}{3}} + C$

$I = \frac{3}{4} {\left({x}^{2} - 1\right)}^{\frac{2}{3}} + C$

Hopefully this helps!