Question

Evaluate: \(\int\frac{2x - 1}{(x^2 - 4)(x + 1)}dx\)

Answer 1

\(\int\frac{2x - 1}{(x^2 - 4)(x + 1)}dx\)

Let

\(\int\frac{(2x - 1)dx}{(x - 2)(x + 2)(x + 1)}\) = \(\int\frac{A}{x - 2} + \frac{B}{x + 2} + \frac{C}{x + 1}dx .....(1)\)

2x – 1 = A(x + 2) (x + 1) +B(x – 2) (x + 1) + c(x – 2)(x + 2) 

put x = 2 4 – 1 = A(4) (3) + B(0) + C(0) 

3 = 12A ⇒ A = \(\frac{1}{4}\)

put x = 2 -4 -1 = A(0) + B(-2) (-2 +1) + C(0) 

-5 = 4B ⇒ B = \(\frac{-5}{36}\)

put x = -1 -2 -1 = A(0) + B(0) + C(-3)1 

-3 = -3c ⇒ c = 1