Write out the form of the partial fraction decomposition of the function appearing in the integral: ∫(−7𝑥−181)/(x^2+9𝑥−22) 𝑑𝑥

Question

Determine the numerical values of the coefficients A and B where 𝐴≤𝐵 and ∫(−7𝑥−181)/(x^2+9𝑥−22) 𝑑𝑥 = A/denominator + B/denominator

 

Answer 1

x^2 + 9x - 22 = (x + 11 )(x - 2 )

(-7x-181) / [(x+11)(x-2)] = A/(x+11) + B/(x-2)

(-7x-181) = a(x-2) + b(x+11)

= (a+b)x + (11b-2a)

a + b = -7

11b - 2a = -181

2*eq1 + eq2:

13b = -195

b = -15

then a = 8

check:

8(x+11) - 15/(x-2)

When these fractions are combined,

8(x-2) - 15(x+11) is the numerator

which equals 8x-16 - 15x - 165 =

-7x - 181

The official p.f.d is

8(x+11) - 15/(x-2)