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Thread: Please show me the steps to this integration...

  1. #1
    tDM
    tDM is offline
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    Angry Please show me the steps to this integration...

    Could somebody please walk me through this integration, thank-you.
    $\displaystyle \int{\alpha^\beta x^{-\beta-1}d\beta}$
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  2. #2
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    Hello, tDM!

    $\displaystyle \int{\alpha^\beta x^{-\beta-1}d\beta}$
    The variables are very confusing . . . I'll modify the problem.

    . . $\displaystyle \int a^xb^{-x-1}\,dx$ . . .There! ... $\displaystyle a$ and $\displaystyle b$ are constants, $\displaystyle x$ is the variable.


    We have: .$\displaystyle a^xb^{-(x+1)} \:=\:\frac{a^x}{b^{x+1}} \;=\;\frac{a^x}{b\cdot b^x} \;=\;\frac{1}{b}\cdot\frac{a^x}{b^x} \;=\;\frac{1}{b}\left(\frac{a}{b}\right)^x $


    Therefore: .$\displaystyle \frac{1}{b}\int\left(\frac{a}{b}\right)^xdx \;=\;\frac{1}{b}\cdot\frac{\left(\frac{a}{b}\right )^x}{\ln\left(\frac{a}{b}\right)} + C \;=\;\frac{1}{b\ln(\frac{a}{b})}\cdot\left(\frac{a }{b}\right)^x + C$

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