express-in-partial-fraction-3x-2-x-2-1-x-1-

Question Number 10907 by j.masanja06@gmail.com last updated on 01/Mar/17

express in partial fraction

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

Answered by sandy_suhendra last updated on 01/Mar/17

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

let \frac{a}{(x + 1)} + \frac{b}{{(x + 1)}^{2}} + \frac{c}{(x - 1)} = \frac{3 x + 2}{{(x + 1)}^{2} (x - 1)}

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

\frac{{ax}^{2} - a + bx - b + {cx}^{2} + 2 cx + c}{{(x + 1)}^{2} (x - 1)} = \frac{3 x + 2}{{(x + 1)}^{2} (x - 1)}

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

a + c = 0 \Rightarrow a = - c

b + 2 c = 3 \Rightarrow b = 3 - 2 c

- a - b + c = 2 \Rightarrow c - 3 + 2 c + c = 2 \Rightarrow c = \frac{5}{4}

a = - \frac{5}{4}

b = 3 - 2 (\frac{5}{4}) = \frac{1}{2}

the partial fraction :

\frac{- 5}{4 (x + 1)} + \frac{1}{2 {(x + 1)}^{2}} + \frac{5}{4 (x - 1)}