Question

$\frac{x + 1}{(x - 1)(x - 2)(x - 3)} =$

• A. $\frac{1}{x - 1} + \frac{3}{x - 2} + \frac{1}{x - 3}$
• B. $- \frac{3}{x - 1} + \frac{1}{x - 2} + \frac{2}{x - 3}$
• C. $\frac{1}{x - 1} - \frac{3}{x - 2} + \frac{2}{x - 3}$
• D. None of these

Answer 1

Answer: (C)

$\frac{1}{x - 1} - \frac{3}{x - 2} + \frac{2}{x - 3}$

Answer 2

$\frac{x + 1}{(x - 1)(x - 2)(x - 3)} = \frac{A}{x - 1} + \frac{B}{x - 2} + \frac{C}{x - 3}$

$\Rightarrow x + 1 = A(x - 2)(x - 3) + B(x - 1)(x - 3) + C(x - 1)(x - 2)$

Putting $x = 1,A = 1$; $\frac{x}{1 + x} - \log(1 + x) = 1 - \frac{1}{1 + x} - \log(1 + x)$ gives $B = - 3$,

For $x = 3,C = 2$

$\therefore$Given expression = $\frac{1}{x - 1} - \frac{3}{x - 2} + \frac{2}{x - 3}$.