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Math Help - Integration of Exponential Functions

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    Integration of Exponential Functions

    I need help integrating a math problem!!!

    2^x/(1+2^x) dx
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  2. #2
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    Quote Originally Posted by sarah s View Post
    I need help integrating a math problem!!!

    2^x/(1+2^x) dx
    Hello Sarah,

    use substitution:

    u(x)=1+2^x=1+e^{\ln(2) \cdot x}\ \Longrightarrow\ \frac{du}{dx}=\ln(2) \cdot e^{\ln(2) \cdot x}=\ln(2) \cdot 2^x. That means:

    du=\ln(2) \cdot 2^x \cdot dx

    With

    \int{\frac{2^x}{1+2^x}}dx=\frac{1}{\ln(2)}\int{\fr  ac{\ln(2) \cdot 2^x}{1+2^x}}dx=\frac{1}{\ln(2)} \int{\frac{1}{u}}du

    \frac{1}{\ln(2)} \int{\frac{1}{u}}du=\frac{1}{\ln(2)} \ln(u)+C. Now re-substitute:

    \int{\frac{2^x}{1+2^x}}dx=\frac{1}{\ln(2)} \cdot \ln(1+2^x)+C

    EB
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