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

Jun 27, 2018

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

#### Explanation:

Integrate by parts:

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

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

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

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

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

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

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