# How do you integrate ln(x) x^ (3/2) dx?

May 6, 2016

$\int \ln \left(x\right) {x}^{\frac{3}{2}} \mathrm{dx} = \frac{2}{5} {x}^{\frac{5}{2}} \left(\ln \left(x\right) - \frac{2}{5}\right) + C$

#### Explanation:

For this problem, we will use integration by parts:

Let $u = \ln \left(x\right)$ and $\mathrm{dv} = {x}^{\frac{3}{2}} \mathrm{dx}$

Then $\mathrm{du} = \frac{1}{x} \mathrm{dx}$ and $v = \frac{2}{5} {x}^{\frac{5}{2}}$

Applying the formula $\int u \mathrm{dv} = u v - \int v \mathrm{du}$ gives us

$\int \ln \left(x\right) {x}^{\frac{3}{2}} \mathrm{dx} = \frac{2}{5} {x}^{\frac{5}{2}} \ln \left(x\right) - \int \frac{2}{5} {x}^{\frac{5}{2}} \cdot \frac{1}{x} \mathrm{dx}$

$= \frac{2}{5} {x}^{\frac{5}{2}} \ln \left(x\right) - \frac{2}{5} \int {x}^{\frac{3}{2}} \mathrm{dx}$

$= \frac{2}{5} {x}^{\frac{5}{2}} \ln \left(x\right) - \frac{2}{5} \left(\frac{2}{5} {x}^{\frac{5}{2}}\right) + C$

$= \frac{2}{5} {x}^{\frac{5}{2}} \left(\ln \left(x\right) - \frac{2}{5}\right) + C$