# How do you find the integral ln x / x?

Sep 12, 2015

$\int \ln \frac{x}{x} = {\left(\ln x\right)}^{2} / 2 + C$

#### Explanation:

int ln x / x = ?

This is a classic application of u -substitution.

Let $u = \ln x$. Then $\mathrm{du} = \frac{1}{x} \mathrm{dx}$.

Now we have

$\int \ln \frac{x}{x} \mathrm{dx} = \int u \mathrm{du}$

We can evaluate this easily by using the power rule:

$\int u \mathrm{du} = {u}^{2} / 2 + C$

Now, substituting back $u$, we find

${u}^{2} / 2 + C = {\left(\ln x\right)}^{2} / 2 + C$

And there is our solution.

$\int \ln \frac{x}{x} = {\left(\ln x\right)}^{2} / 2 + C$