Question

Розкласти на суму простих дробів

Answer 1

$\frac{x}{ x^{3}-1 } = \frac{x}{(x-1)( x^{2} +x+1)} = \frac{A}{x-1} + \frac{Bx+C}{x^{2} +x+1} \\ \\ \frac{A(x^{2} +x+1)+(Bx+C)(x-1)}{(x-1)( x^{2} +x+1)} \\ \\ x = A(x2+x+1) + (Bx + C)(x-1) \\ x^{2} : A + B = 0 \\ x: A -B + C = 1 \\ 1: A -C = 0 \\ A = 1/3;B = -1/3;C = 1/3 \\ \\ \frac{x}{(x-1)( x^{2} +x+1)}= \frac{ \frac{1}{3} }{x-1}+ \frac{ \frac{-1}{3} x+ \frac{1}{3} }{x^{2} +x+1}$