How to Integrate ln(x)

So how do we integrate ln(x)dx? This is actually a very easy but interesting integral, and the counter intuitive thing is, is we have to use integration by parts.

Here's the rule for integration by parts:

\int u\mathrm{d}v=uv-\int v\mathrm{d}u

Now you may be wondering how would this work, considering there is only one partsin this integral. Actually, there are two parts:

We can used the ln(x) as one part and dx as the other part.

So if we let...

\begin{matrix} u=\ln x; & \mathrm{d}v=\mathrm{d}x \end{matrix}

Then it follows that...

\begin{matrix} \mathrm{d}u=\frac{1}{x} \mathrm{d}x; & v=x \end{matrix}

Now that we have these, let's substitute them into the rule...

\int \ln x \mathrm{d}x=\ln x\cdot x-\int \frac{1}{x}\mathrm{d}x

Please watch the video for the full tutorial!