Question

Write out the form of the partial fraction decomposition of the function appearing in the integral: ∫−4x−34x2+2x−8dx ∫−4x−34x2+2x−8dx Determine the numerical values of the coefficients, AA and BB, where A≤BA≤B and −4x−34x2+2x−8 =Adenominator+Bdenominator.

A = ____.
B= _____.

Answer 1

Looks like the integrand is

`(-4x-34)/(x^2+2x-8)`

The denominator is factorized as

`x^2+2x-8=(x+4)(x-2)`

Then we want to find constants
`A,B` such that

`-(4x+34)/(x^2+2x-8)=\frac A{x+4}+\frac B{x-2}`

`\implies-4x-34=A(x-2)+B(x+4)`

We can use the cover-up method to easily find
`A` and
`B`:

• If
  `x=-4`, then
  `-18=-6A\implies A=3`
• If
  `x=2`, then
  `-42=6B\implies B=-7[tex]</li></ul><p>so that</p><p>[tex]-(4x+34)/(x^2+2x-8)=\frac3{x+4}-\frac7{x-2}`