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!

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Answered by Edward C. Maths tutor

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