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Math Help - Integration of Exponential Functions

  1. #1
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    Integration of Exponential Functions

    HELP!!! I have a homework problem.....i'm not sure where to begin with this one. I've tried many different ways and still cannot seem to get a correct answer.


    2^x/(1+2^x) dx
    Last edited by sarah s; January 11th 2007 at 09:33 PM.
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  2. #2
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    Hello, Sarah!

    You're expected to know that the derivative of 2^x is: . 2^x\!\cdot\ln 2


    \int \frac{2^x}{1+2^x}\,dx

    Let u \:=\:1 + 2^x\quad\Rightarrow\quad du = 2^x\ln 2\,dx\quad\Rightarrow\quad dx = \frac{du}{2^x\ln 2}

    Substitute: . \int\frac{2^x}{u}\left(\frac{du}{2^x\ln 2}\right) \;=\;\frac{1}{\ln 2}\int\frac{du}{u}\;=\;\frac{1}{\ln 2}\!\cdot\ln u + C

    Back-substitute: . \frac{1}{\ln 2}\!\cdot\ln(1 + 2^x) + C

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