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Evaluate: int(x^2+1)/((x-1)^2(x+3))\ dx...

Evaluate: `int(x^2+1)/((x-1)^2(x+3))\ dx`

Text Solution

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`int frac{x^{2}-1}{(x-1)^{2}(x+3)} d x`

`=int frac{(x-1)(x+1)}{(x-1)^{2}(x+3)} d x`

`=int frac{x+1}{(x-1)(x+3)} d x`

`=int frac{2 x+2}{2(x-1)(x+3)} d x`

`=int frac{(x+3)+(x-1)}{2(x-1)(x+3)} d x`

`=int frac{x+3}{2(x-1)(x+3)} d x+int frac{1}{2(x-1)(x+3)} d x`

`=int frac{1}{2(x-1)} d x+int frac{1}{2(x+3)} d x`

`=frac{1}{2} times[ln (x-1)+ln (x+3)+c]`, where c is a constant

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