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 $0<k<1$

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 $0<k<1$

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.