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Integration by Substitution

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1. I got parts a & b but there's something I want to ask in general but using this as an example. In part c) I have to do dx/du but how do I know when to do that? I'm sure it hasn't come up in all the papers but I tripped up on the question because I got up to :

4/2 (b:a)[x^3(lnu)du] but didn't get past this stage till I went through it with Exam Solutions, can anyone tell me when to look out for it please
2. You have

and you need to change this to an integral du using the substitution

The way to change dx to du is to find du/dx:

Now substutute everything in, remembering that .

Post your working if you're still stuck. Or post the specific part that confuses you.
3. You have to do dx/du (or similar) for every substitution. It is how you change the dx into a du.
4. Thank you both, I got it now

I was getting confused before because I thought it was an additional step. One question though - when you're finding the area/volume of a curve using Integration and I know when they are parametric, you add a little bit after the integral of y^2 as in it'll dx/dt*dt or something?

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