Question

How do you express `1/(x^3 +4x)` in partial fractions?

Answer 1

let
`1/(x^3+4x)=1/(x(x^2+4))=a/x+(bx)/(x^2+4)=(a(x^2+4)+bx^2)/(x(x^2+4))`
Now
`1=(a(x^2+4)+bx^2)`
putting x=0 in the above identity we have `4a=1 =>a=1/4`
Again putting x=1, we have `5a+b=1`
putting `a=1/4` in the 2nd equation `5a+b=1`
we get `5/4+b=1=>b=-1/4`

Hence we can write
`1/(x^3+4x)=1/4(1/x-x/(x^2+4))`