# What is the integral of ln(sqrt(x))?

May 2, 2015

By part :

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

$\mathrm{du} = 1$
$u = x$

$v = \ln \left(\sqrt{x}\right)$
$\mathrm{dv} = \frac{1}{2 x}$

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

$\left[x \ln \left(\sqrt{x}\right) - \frac{1}{2} x\right]$

don't forget $\ln \left({a}^{b}\right) = b \ln \left(a\right)$

$\left[\frac{1}{2} x \ln \left(x\right) - \frac{1}{2} x\right]$

factorize by $\frac{1}{2} x$ and don't forget the constant !

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