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$\int \frac{e^x}{\left(2+e^x\right)\left(e^x+1\right)} d x=$
(where $C$ is $a$ constant of integration.)
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(where $C$ is $a$ constant of integration.)
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Verified Answer
The correct answer is:
$\log \left(\frac{e^x+1}{e^x+2}\right)+C$
$\begin{aligned} & \int \frac{e^x}{\left(2+e^x\right)\left(e^x+1\right)} d x \\ & =\int \frac{d t}{(t+2)(t+1)}=\int\left(\frac{-1}{t+2}+\frac{1}{t+1}\right) d t \\ & =-\log |t+2|+\log |t+1|+c \\ & =\log \left|\frac{t+1}{t+2}\right|+c \\ & =\log \left|\frac{e^x+1}{e^x+2}\right|+c\end{aligned}$
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