Convergence of an Improper Integral Involving Exponential Functions

[SirPlus]

Homework Statement

1.Determine the divergence/convergence of the following improper integrals by the evaluation of the limit:

$\int_{0}^{∞} \frac{dx}{e^{-x} + e^{x}}$

Homework Equations

The Attempt at a Solution

Let u = e^x
∴ du = e^x dx

I ended up with:

$\int_{0}^{∞} \frac{u du}{u^{2} + 1}$
I have no idea on how to integrate the above ...

 

[vanhees71]

substitute $v=u^2$.

 

[tiny-tim]

Hi SirPlus!

$\int_{0}^{∞} \frac{dx}{e^{-x} + e^{x}}$

Let u = e^x
∴ du = e^x dx

fine so far

$\int_{0}^{∞} \frac{u du}{u^{2} + 1}$

Sorry, but that's completely wrong (including the limits).

Start again, and this time write it out carefully step-by-step, with no short-cuts!