Partial Fraction Assistance

Apr 2009
11
0
How would you find the partial fraction decomposition of:
[-e^-x] / [x*(x+1)]

Using
[A*(x+1)] - [B*(x)] = [-e^-x]
I manage to get
[-1/x] - [e/(x+1)]

just doesnt seem right, could someone confirm or point out where im going wrong.
 

Opalg

MHF Hall of Honor
Aug 2007
4,039
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Leeds, UK
How would you find the partial fraction decomposition of:
[-e^-x] / [x*(x+1)]

Using
[A*(x+1)] - [B*(x)] = [-e^-x]
I manage to get
[-1/x] - [e/(x+1)]

just doesnt seem right, could someone confirm or point out where im going wrong.
You can only use the usual rule for finding A and B when the numerator of the original fraction is of the form cx+d (where c and d are constants). In this case, numerator is \(\displaystyle -e^{-x}\), which is obviously not of that form. So you have to extract that numerator, and write the fraction as \(\displaystyle -e^{-x}\frac1{x(x+1)}\). Then apply the usual partial fractions procedure to get \(\displaystyle \frac1{x(x+1)} = \frac1x - \frac1{x+1}\). Finally, bring back the factor \(\displaystyle -e^{-x}\) to get \(\displaystyle \frac{-e^{-x}}{x(x+1)} = \frac{-e^{-x}}x - \frac{-e^{-x}}{x+1}\).