Convergence of Integral: How to Prove for 0<k<1?

[rioo]

Homework Statement

Show that \int^{\infty}_{-\infty} \frac{e^{kx}}{1+e^{x}}dx converges if 0&lt;k&lt;1

Homework Equations

None

The Attempt at a Solution

Well if I can show that the integral is dominated by another that converges then I'm done, but I haven't been able to come up with one. I've tried manipulating the integrand (moving the e^{kx} to the bottom and checking limits. The integrand does go to zero at -\infty \mathrm{and\ } \infty, but that doesn't guarantee convergence...

 

[LCKurtz]

Homework Statement

Show that \int^{\infty}_{-\infty} \frac{e^{kx}}{1+e^{x}}dx converges if 0&lt;k&lt;1

Homework Equations

None

The Attempt at a Solution

Well if I can show that the integral is dominated by another that converges then I'm done, but I haven't been able to come up with one. I've tried manipulating the integrand (moving the e^{kx} to the bottom and checking limits. The integrand does go to zero at -\infty \mathrm{and\ } \infty, but that doesn't guarantee convergence...

Look at the two cases $\int_0^\infty$ and $\int_{-\infty}^0$ separately and use different overestimates on the different intervals. If you can show they are both finite you are done.