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

1 Answer
Dec 16, 2015

Solve to find:

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

Explanation:

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))