# How do you show whether the improper integral int e^x/ (e^2x+3)dx converges or diverges from 0 to infinity?

Oct 25, 2015

Assuming that the intended integral is $\int {e}^{x} / \left({e}^{2 x} + 3\right) \mathrm{dx}$ see below.

#### Explanation:

Integration by substitution will get an arctan whose argument involves ${e}^{x}$. As $x \rightarrow \infty$, the argument will $\rightarrow \infty$, so arctan $\rightarrow \frac{\pi}{2}$

Let $u = {e}^{3}$ so $\mathrm{du} = {e}^{x} \mathrm{dx}$ and the integral becomes

$\int \frac{1}{{u}^{2} + 3} \mathrm{du}$

Now let $u = \sqrt{3} t$ to get

$\frac{1}{\sqrt{3}} \int \frac{1}{{t}^{2} + 1} \mathrm{dt} = \frac{1}{\sqrt{3}} {\tan}^{-} 1 \left(t\right)$

Where $t = \frac{u}{\sqrt{3}} = {e}^{x} / \sqrt{3}$

So $\int {e}^{x} / \left({e}^{2 x} + 3\right) \mathrm{dx} = \frac{1}{\sqrt{3}} {\tan}^{-} 1 \left({e}^{x} / \sqrt{3}\right)$

I have omitted the details to properly express the calculation of an improper integral. I assume that the indefinite integral was the difficulty.