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Integration by Substitution

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    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
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    You have

     \displaystyle \int x^3 \ln (x^2+2) \ dx

    and you need to change this to an integral du using the substitution \displaystyle u=x^2+2

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

     \displaystyle \frac{du}{dx} = 2x \implies dx = \frac{du}{2x}

    Now substutute everything in, remembering that x^2=u-2.

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