Write a value of $int rac{1}{1+2 mathrm{e}^{mathrm{x}}} mathrm{dx}$

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.