Determin wheter the following integral is convergent and evaluate it if convergent.?

Question

0 to infinity integral sign.....e^x / (e^2x + 3) dx

(0 is the bottom and infinity is at the top)

please show work so I could understand. Thank you

Answer 1

You can do it by comparison.
e^x / (e^2x + 3) is less than e^x /e^2x which is equal to 1/e^x and since the integral of 1/e^x is convergent so is your function.

Answer 2

Let´s call e^x= z so e^x dx = dz so we have the integral of
dz/(z^2+3) = 1/3 dz/[(z/sqrt3)^2+1]= 1/3 Arctan (z/sqrt3)*sqrt3
taken for z between 1 and infinity
= 1/3 sqrt3*pi/2