Integration by partial fractions help?

I'm trying to find the integral of

(-12e^x-20)/(e^(2x)+6e^x+5)

using integration by partial fractions, but I keep getting stuck at

-4 (/int/ (1/2)/(e^x+1) + /int/ (5/2)/(e^x+5))

I'm not sure if I messed up on one of the intermediate steps. If not, how do I proceed from here to finish integrating?

Note: the answer is supposed to be -4x+2ln(1+e^x)+2ln(5+e^x), but I have no idea how to get there.

Any help would be appreciated, thank you!

Re: Integration by partial fractions help?

Quote:

Originally Posted by

**jiel** I'm trying to find the integral of

(-12e^x-20)/(e^(2x)+6e^x+5)

using integration by partial fractions, but I keep getting stuck at

-4 (/int/ (1/2)/(e^x+1) + /int/ (5/2)/(e^x+5))

I'm not sure if I messed up on one of the intermediate steps. If not, how do I proceed from here to finish integrating?

Note: the answer is supposed to be -4x+2ln(1+e^x)+2ln(5+e^x), but I have no idea how to get there.

Any help would be appreciated, thank you!

$f(x)=\dfrac{-12 e^x - 20}{e^{2x}+6e^x+5}$

let $u=e^x$

$f(u)=\dfrac{-12u-20}{u^2+6u+5}$

$f(u)=\dfrac{-12u-20}{(u+5)(u+1)}$

$f(u) = \dfrac A {u+5} + \dfrac B {u+1}$

$A(u+1) + B(u+5) = -12u - 20$

You should be able to finish this from here. Just substitute back $e^x$ for $u$ when you are finished.

Re: Integration by partial fractions help?