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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= _____.

Respuesta :

LammettHash

Looks like the integrand is

[tex]\dfrac{-4x-34}{x^2+2x-8}[/tex]

The denominator is factorized as

[tex]x^2+2x-8=(x+4)(x-2)[/tex]

Then we want to find constants [tex]A,B[/tex] such that

[tex]-\dfrac{4x+34}{x^2+2x-8}=\dfrac A{x+4}+\dfrac B{x-2}[/tex]

[tex]\implies-4x-34=A(x-2)+B(x+4)[/tex]

We can use the cover-up method to easily find [tex]A[/tex] and [tex]B[/tex]:

  • If [tex]x=-4[/tex], then [tex]-18=-6A\implies A=3[/tex]
  • If [tex]x=2[/tex], then [tex]-42=6B\implies B=-7[tex]

so that

[tex]-\dfrac{4x+34}{x^2+2x-8}=\dfrac3{x+4}-\dfrac7{x-2}[/tex]

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