How to integrate a fraction of sums of exponentials?

[DRJP]

Is it possible to have an solution to this sort of integral? And if not, why not?

\int_0^\infty \frac{e^{-ax}}{e^{-bx}+e^{-cx}}dx

Is a Taylor expansion the only way forward?

Many thanks
David

 

[dextercioby]

Is it possible to have an solution to this sort of integral? And if not, why not?

\int_0^\infty \frac{e^{-ax}}{e^{-bx}+e^{-cx}}dx

Is a Taylor expansion the only way forward?

Many thanks
David

Use [tex ] instead of inline tex if you're not writing a formula on the same line with words.

\int_0^\infty \frac{e^{-ax}}{e^{-bx}+e^{-cx}}dx

looks better and is easier to read.

As for your question, before jumping to series expansions and substitutions, specify if the arbitrary constants are positive or negative. This makes a huge difference on the final result.
Then try to get rid of as many exponentials as possible. You can make the substitution (a,b,c >0) \displaystyle{e^{-ax}} = t and see what you get.