Question:
Write a value of $\int \frac{1}{1+2 \mathrm{e}^{\mathrm{x}}} \mathrm{dx}$
Solution:
Take $e^{x}$ out from the denominator.
$y=\int \frac{1}{e^{x}\left(e^{-x}+2\right)} d x$
$y=\int \frac{e^{-x}}{\left(e^{-x}+2\right)} d x$
Let, $e^{-x}+2=t$
Differentiating both sides with respect to $x$
$\frac{d t}{d x}=-e^{-x}$
$\Rightarrow-d t=e^{-x} d x$
$y=\int \frac{-d t}{t}$
Use formula $\int \frac{1}{t} d t=\ln t$
$Y=-\ln t+c$
Again, put $e^{-x}+2=t$
$Y=-\ln \left(e^{-x}+2\right)+c$
Note: Don't forget to replace $t$ with the function of $x$ at the end of solution. Always put constant c with indefinite integral.
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