# What is the antiderivative of ln(x)/sqrtx ?

Dec 26, 2015

$2 \sqrt{x} \left(\ln \left(x\right) - 2\right) + C$

#### Explanation:

For the given function, finding the antiderivative is equivalent to finding the indefinite integral. We will proceed by applying Integration by Parts.

Let $u = \ln \left(x\right)$ and $\mathrm{dv} = \frac{1}{\sqrt{x}} \mathrm{dx}$

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

From the integration by parts formula $\int u \mathrm{dv} = u v - \int v \mathrm{du}$

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

$= 2 \sqrt{x} \ln \left(x\right) - 2 \int \frac{1}{\sqrt{x}} \mathrm{dx}$

$= 2 \sqrt{x} \ln \left(x\right) - 2 \left(2 \sqrt{x}\right) + C$

$= 2 \sqrt{x} \left(\ln \left(x\right) - 2\right) + C$