Find the integral of ln(x)

To solve this, we must use integration by parts as we can’t solve it directly. The formula for integration by parts is integral(UdV)=UV-integral(V*dU). The trick with this is to set dV=1 and to set U=ln(x). These multiplied together make ln(x) so the formula is suitable. We first look at working out the variables used in the RHS of the formula. To find V we integrate dV=1 This integrated gives us V=x. We also need to work out dU from U=ln(x). To find this we differentiate U giving dU=1/x.

Now we have everything we need to substitute these values into the formula, we start by working out the individual parts of the formula

Firstly: U*V=ln(x)*x

Secondly: integral(VdU)=integral(x1/x)=integral(1)=x+C (don’t forget the constant of integration)

So overall this gives:

integral(UdV)=UV-integral(V*dU)

integral(ln(x))=x*ln(x)-x+C