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Math Help - Integration by substitution

  1. #1
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    Integration by substitution

    Hi

    I have this expression but don't know how the answer was reached.

    \int\frac{e^{a/x}}{x^{2}}=-\frac{e^{a/x}}{a}

    I'm sure substitution is involved, this is a weak area for me.

    Any help would be greatly appreciated.

    Thanks
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  2. #2
    MHF Contributor chisigma's Avatar
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    The derivative of e^{g(x)} is e^{g(x)}\cdot g^{'}(x)... all right?... and which is the derivative of \frac{a}{x}?...

    Kind regards

    \chi \sigma
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  3. #3
    MHF Contributor red_dog's Avatar
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    Use the substitution \frac{a}{x}=t.

    \frac{1}{x}=\frac{t}{a}\Rightarrow-\frac{1}{x^2}dx=\frac{1}{a}dt\Rightarrow \frac{1}{x^2}dx=-\frac{1}{a}dt.

    Then \int e^t\left(-\frac{1}{a}\right)dt=-\frac{1}{a}\int e^tdt=-\frac{e^t}{a}+C

    Now replace back t with \frac{a}{x}.
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