Ok, I give up, I have tried with $u$-substitution and integration by parts but I can't solve it. The integral is:
$$\int{\frac{e^x dx}{1+e^{2x}}}$$
I have tried $u=e^x$, $u=e^{2x}$ and also integration by parts but I can't solve it. The result should be:
$$\arctan(e^x)$$
Best Answer
Use $u = e^x, du = e^x dx.$
Then you have:
$$\int \frac{du}{1 + u^2} \text{because} \space (e^x)^2 = e^{2x} $$
$$\arctan (u) + C$$
$$\arctan(e^x) + C$$