# How do you integrate int ln(x)/x dx using integration by parts?

$\int \ln \frac{x}{x} \mathrm{dx} = \ln {\left(x\right)}^{2} / 4$
Integration by parts is a bad idea here, you will constantly have $\int \ln \frac{x}{x} \mathrm{dx}$ somewhere. It is better to change the variable here because we know that the derivative of $\ln \left(x\right)$ is $\frac{1}{x}$.
We say that $u \left(x\right) = \ln \left(x\right)$, it implies that $\mathrm{du} = \frac{1}{x} \mathrm{dx}$. We now have to integrate $\int u \mathrm{du}$.
$\int u \mathrm{du} = {u}^{2} / 2$ so $\int \ln \frac{x}{x} \mathrm{dx} = \ln {\left(x\right)}^{2} / 2$