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

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Answer 1 · 2015-10-25T18:31:41

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

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.

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