\(\int\frac{5}{(x^2 + 6x + 9)(x - 3)}dx\)
Let
\(\frac{5}{(x + 3)^2(x - 3)} = \frac{A}{x + 3} + \frac{B}{(x + 3)^2} + \frac{C}{x - 3} ....(1)\)
5 = A(x + 3) (x – 3) + B(x – 3) + c(x + 3)2
put x = 3, 5 = A(0) + B(0) + C(3 +3)2 = 5 = 36C ⇒ c = \(\frac{5}{35}\)
put x = -3, 5 = (A) (0) + B(-3-3) + C(0)2
5 = -6B ⇒C = \(\frac{5}{36}\)
put x = -3, 5 = (A) (0) + B(-3-3) + C(0)2
5 = -6B ⇒ B = \(\frac{5}{-6}\)
Comparing coefficients of ×2 both sides we get
0 = A + C ⇒ A = -C = \(\frac{-5}{36}\)
