Integration of $\int \frac{e^x}{e^{2x} + 1}dx$

Question

I came across this question and I was unable to solve it. I know a bit about integrating linear functions, but I don't know how to integrate when two functions are divided. Please explain. I'm new to calculus.

Question: $$\int \frac{e^x}{e^{2x} + 1}dx$$

Thanks in advance.

Answer 1

You may write $$ \int \frac{e^x}{e^{2x} + 1}dx=\int \frac{d(e^x)}{(e^{x})^2 + 1}=\arctan (e^x)+C $$ since $$ \int \frac{1}{u^2 + 1}du=\arctan u+C. $$