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

Jan 9, 2017

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

#### Explanation:

We can calculate the integral by parts:

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

Solving this last integral:

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

Putting it together:

$\int \frac{\ln x}{x} ^ \left(\frac{1}{2}\right) \mathrm{dx} = 2 {x}^{\frac{1}{2}} \ln x - 4 {x}^{\frac{1}{2}} + C = 2 {x}^{\frac{1}{2}} \left(\ln x - 2\right) + C$