Question

1. $int\frac{e^{x}}{7 + e^{2x}}dx$

Explanation:

Step1: Use substitution

Let $u = e^{x}$, then $du=e^{x}dx$. The integral becomes $\int\frac{du}{7 + u^{2}}$.

Step2: Recall integral formula

We know that $\int\frac{1}{a^{2}+x^{2}}dx=\frac{1}{a}\arctan(\frac{x}{a})+C$. Here $a = \sqrt{7}$ and $x = u$. So $\int\frac{du}{7 + u^{2}}=\frac{1}{\sqrt{7}}\arctan(\frac{u}{\sqrt{7}})+C$.

Step3: Substitute back

Substitute $u = e^{x}$ back into the result. We get $\frac{1}{\sqrt{7}}\arctan(\frac{e^{x}}{\sqrt{7}})+C$.

Answer:

$\frac{1}{\sqrt{7}}\arctan(\frac{e^{x}}{\sqrt{7}})+C$