# Partial Fraction Assistance

#### john1985

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
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 $$\displaystyle \frac{-e^{-x}}{x(x+1)} = \frac{-e^{-x}}x - \frac{-e^{-x}}{x+1}$$.