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Evaluate: intx/((x-1)^2(x+2))\ dx...

Evaluate: `intx/((x-1)^2(x+2))\ dx`

Text Solution

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`I=int frac{x d x}{(x-1)^{2}(x+2)}`
Let `frac{x}{(x-1)^{2}(x+2)}=frac{A}{x-1}+frac{B}{(x-1)^{2}}+frac{C}{x+2}`
`Rightarrow frac{x}{(x-1)^{2}(x+2)}=frac{A(x-1)(x+2)+B(x+2)+C(x-1)^{2}}{(x-1)^{2}(x+2)} `
`Rightarrow x=A(x^{2}+2 x-x-2)+B(x+2)+C(x^{2}-2 x+1) `
`Rightarrow x=A(x^{2}+x-2)+B(x+2)+C(x^{2}-2 x+1) `
`Rightarrow x=(A+C) x^{2}+(A+B-2 C) x+(-2 A+2 B+C)`
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