How do you express #(x^(2) - 1) / (x(x^(2)+1))# in partial fractions?

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Answer 1 · 2016-11-26T11:18:52

The answer is #=-1/(x)+(2x)/(x^2+1)#

Let's do the decomposition into partial fractions

#(x^2-1)/(x(x^2+1))=A/(x)+(Bx+C)/(x^2+1)#

#=(A(x^2+1)+x(Bx+C))/(x(x^2+1))#

Therefore,

#x^2-1=A(x^2+1)+x(Bx+C)#

Let #x=0#, #=># #-1=A#

Coefficients of #x^2#

#1=A+B#, #=>#, #B=1-A=2#

coefficients of #x#

#0=C#

Finally, we have

#(x^2-1)/(x(x^2+1))=-1/(x)+(2x)/(x^2+1)#