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    Hi,

    I'm studying maths at the moment, currently learning C4. I'm finding integration really difficult and one of the things that really confuses me at the moment is that
    I don't understand the difference between integration by inspection and integration using the reverse chain rule.

    I'm also struggling with recognising which rule to use for each question. Does anyone know steps/techniques which will help me with this?

    Thanks


    Forgot to say I have a formula for the chain rule, but it doesn't seem to work for inspection so any formula for inspection around?
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    (Original post by djels013.211)
    Hi,

    I'm studying maths at the moment, currently learning C4. I'm finding integration really difficult and one of the things that really confuses me at the moment is that
    I don't understand the difference between integration by inspection and integration using the reverse chain rule.

    I'm also struggling with recognising which rule to use for each question. Does anyone know steps/techniques which will help me with this?

    Thanks


    Forgot to say I have a formula for the chain rule, but it doesn't seem to work for inspection so any formula for inspection around?
    You use reverse chain rule to integrate expressions of the form g(ax+b), which is a composition of the two functions g(x) and ax+b. All you do is integrate g(x) as usual, stick with the ax+b inside it, then just multiply the whole thing by \frac{1}{a}.

    It just comes to the following statement which is proved via substitution:

    Let \dfrac{d}{dx}G(x) = g(x), then \displaystyle \int g(ax+b) .dx = \frac{1}{a}G(ax+b) +c

    Proof:
    Spoiler:
    Show



    \displaystyle I= \int g(ax+b) .dx and now introduce u=ax+b hence du=a .dx thus dx = \frac{1}{a} .du therefore

    \displaystyle I = \int \frac{1}{a} g(u) .du = \frac{1}{a} \int g(u) .du = \frac{1}{a}G(u) + c = \frac{1}{a}G(ax+b)+c




    Integration by inspection is similar but applies to a wider range of integrals and can only get better once you do more integrals and begin to expect what they might integrate to. So for an integral, you look at it, then you get an idea of what it might integrate to. Then you differentiate your idea to see if you get back to the integrand. If you are off by a constant multiple, then you can adjust by multiplying/dividing your idea by it. If you're off by a function of x, then you should try a different idea.
 
 
 
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