How do I integrate by substitution?

Let's take for example the integral 5x²(x³-4)⁴dx. This is very difficult and not at all nice to expand in terms of x, so we can effectively "create" a new variable, u, and set it to equal x³-4. When we differentiate u, we find 3x². (Would write the maths on the whiteboard). As du/dx = 3x², we can say that du = 3x²dx, which we can see is in the original integral. Therefore, we can sub in du, to get the integral 5/3*(x³-4)⁴du. We can also see that the expression inside the brackets is just u, giving the final integral 5/3*u⁴du. This is much easier to integrate, and gives a much simpler answer. We can then sub in u = x³-4 when we have finished. So essentially, we just sub in a single variable which represents a more complicated function, integrate in terms of this simple variable, and then sub in the complicated function at the end. This makes integration much easier!