# What is the antiderivative of ln(2x)/x^(1/2)?

Feb 9, 2017

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

#### Explanation:

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

We set it up for IBP:
$= \int \ln \left(2 x\right) \frac{d}{\mathrm{dx}} \left(2 {x}^{\frac{1}{2}}\right) \mathrm{dx}$

Applying the IBP:
$= 2 {x}^{\frac{1}{2}} \ln \left(2 x\right) - \int \frac{d}{\mathrm{dx}} \left(\ln \left(2 x\right)\right) \left(2 {x}^{\frac{1}{2}}\right) \mathrm{dx}$

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

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

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

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