How do you write the partial fraction decomposition of the rational expression # 1 / ((x^2 + 1) (x^2 +4))#?

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Answer 1 · 2015-12-16T20:58:14

Solve to find:

#1/((x^2+1)(x^2+4)) = 1/(3(x^2+1)) - 1/(3(x^2+4))#

Neither of the quadratics #(x^2+1)# and #(x^2+4)# have linear factors with Real coefficients, so let's leave them as quadratics and attempt to solve:

#1/((x^2+1)(x^2+4)) = A/(x^2+1) + B/(x^2+4)#

#=(A(x^2+4)+B(x^2+1))/((x^2+1)(x^2+4))#

#=((A+B)x^2+(4A+B))/((x^2+1)(x^2+4))#

Equating coefficients we find:

#A+B = 0#

#4A+B = 1#

Hence #A=1/3# and #B=-1/3#

So:

#1/((x^2+1)(x^2+4)) = 1/(3(x^2+1)) - 1/(3(x^2+4))#

If we allow Complex coefficients, then we find:

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

#1/(x^2+4) = i/(4(x+2i))-i/(4(x-2i))#

Hence:

#1/(3(x^2+1)) - 1/(3(x^2+4))#

#=i/(6(x+i))-i/(6(x-i)) + i/(12(x-2i))-i/(12(x+2i))#