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
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
Hello, Sarah!
You're expected to know that the derivative of $\displaystyle 2^x$ is: .$\displaystyle 2^x\!\cdot\ln 2$
$\displaystyle \int \frac{2^x}{1+2^x}\,dx$
Let $\displaystyle u \:=\:1 + 2^x\quad\Rightarrow\quad du = 2^x\ln 2\,dx\quad\Rightarrow\quad dx = \frac{du}{2^x\ln 2}$
Substitute: .$\displaystyle \int\frac{2^x}{u}\left(\frac{du}{2^x\ln 2}\right) \;=\;\frac{1}{\ln 2}\int\frac{du}{u}\;=\;\frac{1}{\ln 2}\!\cdot\ln u + C$
Back-substitute: .$\displaystyle \frac{1}{\ln 2}\!\cdot\ln(1 + 2^x) + C$