# What is the antiderivative of lnsqrt( 5 x )?

Jan 9, 2016

$I = \frac{x \left(\ln \left(5\right) + \ln \left(x\right) - 1\right)}{2} + c$

#### Explanation:

$I = \int \ln \left(\sqrt{5 x}\right) \mathrm{dx}$

Using log properties

$I = \frac{1}{2} \int \left(\ln \left(5\right) + \ln \left(x\right)\right) \mathrm{dx}$

$I = \frac{x \ln \left(5\right)}{2} + \frac{1}{2} \int \ln \left(x\right) \mathrm{dx}$

For the last integral say $u = \ln \left(x\right)$ so $\mathrm{du} = \frac{1}{x}$ and $\mathrm{dv} = 1$ so $v = x$

$I = \frac{x \ln \left(5\right)}{2} + \frac{1}{2} \left(\ln \left(x\right) x - \int \frac{x}{x} \mathrm{dx}\right)$
$I = \frac{x \left(\ln \left(5\right) + \ln \left(x\right) - 1\right)}{2} + c$