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Math Help - integral with e

  1. #1
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    integral with e

    Hi. I'm stuck on this integral problem:

    \int{\frac{85}{e^{-x}+1}\,dx}

    I keep playing around with substitution and +/- tricks and such like, and I just don't seem to be figuring it out.
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  2. #2
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    Re: integral with e

    Hey infraRed.

    Hint: Try multiplying the integral inside by x/x and use the substitution u = e^(-x) + 1.
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  3. #3
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    Re: integral with e

    Quote Originally Posted by infraRed View Post
    Hi. I'm stuck on this integral problem:

    \int{\frac{85}{e^{-x}+1}\,dx}

    I keep playing around with substitution and +/- tricks and such like, and I just don't seem to be figuring it out.
    \displaystyle \begin{align*} \int{\frac{85}{e^{-x} + 1}\,dx} &= \int{\frac{85e^x}{e^x \left( e^{-x} + 1 \right)} \, dx} \\ &= \int{\frac{85e^x}{1 + e^x}\,dx} \end{align*}

    Now make the substitution \displaystyle \begin{align*} u = e^x \implies du = e^x\,dx \end{align*} and the integral becomes

    \displaystyle \begin{align*} \int{\frac{85e^x}{1 + e^x}\,dx} &= \int{\frac{85}{1 + u}\,du} \\ &= 85\ln{\left| 1 + u \right|} + C \\ &= 85\ln{\left| 1 + e^x \right|} + C \\ &= 85\ln{ \left( 1 + e^x \right) } + C \textrm{ since } 1 + e^x > 0 \textrm{ for all } x \end{align*}
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