# How to integrate a fraction of sums of exponentials?

1. ### DRJP

1
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

Last edited: Dec 21, 2011
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3. ### dextercioby

12,314
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.