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Resolve ((2x)/(x^(3)-1)) into partial fr...

Resolve `((2x)/(x^(3)-1))` into partial fractions:

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We have, `(2x)/(x^(3)-1)=(2x)/((x-1)(x^(2)+x+1))`
So, let `(2x)/((x-1)(x^(2)+x+1))=A/(x-1) + (Bx+C)/(x^(2)+x+1)`
Then, `2x=A(x^(2)+x+1)+(Bx+C)(x-1)`………..(i)
Putting, `x-1=0` or, x=1 in (i), we get 2=3A `rArr A=2/3`
Putting x=0 in (i), we get `A-C=0 rArr C=A=2/3`
Putting `x=-1` in (i), we get `-2 = A+2B-2C rArr -2=2/3 +2B-4/3 rArr B=-2/3`
`therefore (2x)/(x^(3)-1)=2/3.1/(x-1)+((-2//3)x+2//3)/(x^(2)+x+1)` or `(2x)/(x^(3)-1) = 2/31/(x-1) + 2/3(1-x)/(x^(2)+x+1)`
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