Question

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

Answer 1

Assuming that the intended integral is `int e^x/(e^(2x)+3) dx` see below.

Explanation:

Integration by substitution will get an arctan whose argument involves `e^x`. As `xrarroo`, the argument will `rarroo`, so arctan `rarr pi/2`

Let `u = e^3` so `du = e^x dx` and the integral becomes

`int 1/(u^2+3) du`

Now let `u = sqrt3 t` to get

`1/sqrt3 int 1/(t^2+1) dt = 1/sqrt3 tan^-1(t)`

Where `t = u/sqrt3 = e^x/sqrt3`

So `int e^x/(e^(2x)+3) dx = 1/sqrt3 tan^-1 (e^x/sqrt3)`

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