# Integral of xln(x)

In this post I am going to explain step-by-step how to integrate the function *x*ln(*x*). We can solve it by using integration by parts (click here if you want to know more about it), and as you will see something interesting will happen!

Let’s let *u*=*x*, so that *du*=*dx*. If we let *u*=ln(*x*), it means that *dv*=*xdu* and so *v*=*x*²/2, and we don’t want the degree of the *x* to increase. To find *v*, we need to integrate *dv*, and as we saw here, the integral of ln(*x*) is *x*(ln(*x)*-1); therefore:

Now we can multiply *u* and *v* cross-wise and *v* and *du* horizontally:

Let’s apply linearity:

At this point you can notice that we have the integral we had at the beginning; this means

We can move the integral on the right-hand side to the left and treat everything like an equation:

And this is the solution to the integral!

Hope everything was clear, if you need any explanation feel free to leave a comment and I’ll be happy to help you!