Integration of Exponential Functions

**sarah s** · January 11th 2007, 09:17 PM

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

**Soroban** · January 11th 2007, 09:38 PM

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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