Express (frac{1}{x^{3}-1}) in partial fractions

Correct Answer: Option A

Explanation

(frac{1}{x^{3} – 1})

(x^{3} – 1 = (x – 1)(x^{2} + x + 1))

(frac{1}{x^{3} – 1} = frac{A}{x – 1} + frac{Bx + C}{x^{2} + x + 1})

(frac{1}{x^{3} – 1} = frac{A(x^{2} + x + 1) + (Bx + C)(x – 1)}{x^{3} – 1})

Comparing the two sides of the equation,

(A + B = 0 … (1))

(A – B + C = 0 … (2))

(A – C = 1 … (3))

From (3), (C = A – 1), putting that in (2),

(A – B = -C implies A – B = 1 – A)

(2A – B = 1 … (4))

(1) + (4): (3A = 1 implies A = frac{1}{3})

(A = -B implies B = -frac{1}{3})

(C = A – 1 implies C = frac{1}{3} – 1 = -frac{2}{3})

(therefore frac{1}{x^{3} – 1} = frac{1}{3(x – 1)} – frac{x – 2}{3(x^{2} + x + 1)})

= (frac{1}{3}(frac{1}{x – 1} – frac{(x + 2)}{x^{2} + x + 1}))