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!

Answered by Edward C. Maths tutor

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