Question

How do you express `x/((x-1)(x^2+4)` in partial fractions?

Answer 1

The answer is `x/((x-1)(x^2+4))=1/(5(x-1))+(-x+4)/(5(x^2+4))`

Explanation:

Let `x/((x-1)(x^2+4))=A/(x-1)+(Bx+C)/(x^2+4)`
`=(A(x^2+4)+(Bx+C)(x-1))/((x-1)(x^2+4))`
So `x=A(x^2+4)+(Bx+C)(x-1)`
if `x=0` `=>` `0=4A-C`
coefficients of `x^2` `=>` `0=A+B`
coefficents of `x` `=>` `1=-B+C`
Solving for `A,B, C`
`A=1/5`
`B=-1/5`
`C=4/5`

`:.x/((x-1)(x^2+4))=1/(5(x-1))+(-x+4)/(5(x^2+4))`