We are asked to solve the integral $\int \frac{e^x}{4(1+e^{2x})} dx$.

1. Problem Description

We are asked to solve the integral $\int \frac{e^x}{4(1+e^{2x})} dx$.

2. Solution Steps

We can solve this integral using a substitution. Let $u = e^x$. Then $du = e^x dx$.

Substituting these into the integral, we have

$\int \frac{e^x}{4(1+e^{2x})} dx = \int \frac{1}{4(1+u^2)} du = \frac{1}{4} \int \frac{1}{1+u^2} du$.

We know that $\int \frac{1}{1+u^2} du = \arctan(u) + C$.

Therefore, we have

$\frac{1}{4} \int \frac{1}{1+u^2} du = \frac{1}{4} \arctan(u) + C$.

Now, we substitute $u = e^x$ back into the expression:

$\frac{1}{4} \arctan(e^x) + C$.

3. Final Answer

The final answer is $\frac{1}{4}\arctan(e^x) + C$.