Solve this definite integral I have a problem with this definite integral. Please help me:
$$ f(x)= {\int }\frac{1}{1+e^{2x}} dx $$

Put $w=e^{2x}$, then $dw=2e^{2x}dx$, and
$$\int \frac{1}{1+e^{2x}}dx=\int\frac{1+e^{2x}-e^{2x}}{1+e^{2x}}dx=\int dx -\int \frac{e^{2x}}{1+e^{2x}}dx =x-\int \frac{\frac{dw}{2}}{1+w}=$$
$$=x-\frac{1}{2}\int \frac{dw}{w+1}=x-\frac{1}{2}\ln|w+1|+c=x-\frac{1}{2}\ln (1+e^{2x})+c. $$
