The partial fractions form of x3−x+1x2−1 is

x3−x=1x2−1=x(x2−1)+1x2−1 =x+1x2−1 =x+1(x−1)(x+1) =x+12[1x−1−1x+1] ∴x+12(x−1)−12(x+1)

The partial fractions form of x3−x+1x2−1 is

x3−x=1x2−1=x(x2−1)+1x2−1 =x+1x2−1 =x+1(x−1)(x+1) =x+12[1x−1−1x+1] ∴x+12(x−1)−12(x+1)

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The partial fractions form of $\frac{x^{3} - x + 1}{x^{2} - 1}$ is

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x + \frac{1}{2 (x - 1)} + \frac{1}{2 (x + 1)}

x + \frac{1}{2 (x - 1)} - \frac{1}{2 (x + 1)}

x - \frac{1}{2 (x - 1)} + \frac{1}{2 (x + 1)}

x - \frac{1}{2 (x - 1)} - \frac{1}{2 (x + 1)}

answer is B.

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Detailed Solution

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

$= x + \frac{1}{x^{2} - 1}$

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

$= x + \frac{1}{2} [\frac{1}{x - 1} - \frac{1}{x + 1}]$

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

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The partial fractions form of x3−x+1x2−1 is

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The partial fractions form of $\frac{x^{3} - x + 1}{x^{2} - 1}$ is