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Thread: integration problem

  1. #1
    Jul 2009

    Question integration problem

    Hello, my integration problem is as follows:


    where are a and b are constants.

    The solution is given to be: 1/(b-a) * ln[(b-x)/(a-x)]

    I'm unsure of the steps carried out to get this answer. Any help would be greatly appreciated!

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  2. #2
    MHF Contributor
    Oct 2008
    Just in case a picture helps...

    Straight continuous lines differentiate downwards with respect to x, and the straight dashed line with respect to the dashed balloon expression. So the forking network on the right satisfies the chain rule

    Hope this helps, or doesn't further confuse. Step by step in a few minutes... take cover - another pic coming, to show how to differentiate $\displaystyle \frac{b - x}{a - x}$...

    Don't integrate - balloontegrate!

    Balloon Calculus Forum

    Differentiating the inner function...

    In balloon world, differentiating is a process going clockwise from top left. First, identify the expression as a product or composite function - here we have a product shape...

    ... containing within it a chain shape like the one in the first picture...

    (The whole thing is more or less the 'quotient rule for differentiation'.)

    Once you have a chain shape, identify the inner function for the dashed balloon, differentiate that for the right fork (the 'by-product') and differentiate the outer with respect to the inner (travel down the straight dashed line). Then simplify, ending up bottom left.

    Integrating is generally a less sure-footed process, going anti-clockwise from bottom left. In this case (for your original problem) I'm not sure I can offer a clear step-by-step (as opposed to trial and error) reasoning that would have landed us with the successful choice of dashed balloon expression, had we not been told. However, once you know (from differentiating clockwise) that it works, you only need to show that all the links (starting, say, with the lower equals sign) are valid.

    Edit: to clarify...

    You know that a particular choice of dashed balloon expression (= u-sub) works if: you can integrate with respect to it (up the dashed straight line) AND the consequent product on the lower level is one that you need only muliply by some constant in order to make the lower equals sign valid.
    Last edited by tom@ballooncalculus; Jul 7th 2009 at 06:31 AM.
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