\(\int\frac{5x + 7}{(x - 2)^2(x + 3)}dx\)
Let
\(\frac{5x + 7}{(x - 2)^2(x + 3)} = \frac{A}{x - 2} + \frac{B}{(x - 2)^2} + \frac{C}{x + 3}\)
5x + 7 = A(x – 2) (x + 3) + B(x +3) + (x – 2)2
put x = 2, 10 + 7 = A(0) + B(2 + 3) + c(0)2 = 17 = 5B ⇒ B = \(\frac{17}{5}\)
put x = -3 -15 + 7 = A(0) + B(0) + c(-5)2 = -8 = 25c ⇒ c = \(\frac{17}{5}\)
Comparing the coefficient of ×2 both sides we get 0 = A + c ⇒ A = -c = \(\frac{8}{25}\)
