Evaluate the improper integral $\int_{-\infty}^{\infty} \dfrac {e^{x}}{1+e^{2x}} dx$
My Attempt $$=\lim_{a\to -\infty} \int_{a}^{0} \dfrac {e^{x}}{1+e^{2x}} dx + \lim_{b\to \infty} \int_{0}^{b} \dfrac {e^{x}}{1+e^{2x}} dx$$ Substituting $e^x=t$ and evaluating gives the integration as $\tan^{-1} e^{x}$. Then solving the limits I get the value of definite integral as $\dfrac {\pi}{2}$. However the book says the limit doesn't exist. Is there any point where I made an error?