Evaluate: `int(5x^2-1)/(x(x-1)(x+1))\ dx`

Explanation

We have,

`I=int frac{(5 x^{2}-1) d x}{x(x-1)(x+1)}`

Let `frac{5 x^{2}-1}{x(x-1)(x+1)}=frac{A}{x}+frac{B}{x-1}+frac{C}{x+1}`

`Rightarrow frac{5 x^{2}-1}{x(x-1)(x+1)}=frac{A(x^{2}-1)+B x cdot(x+1)+C cdot x}{x(x-1)(x+1)} `

`Rightarrow 5 x^{2}-1=A(x^{2}-1)+B cdot x(x+1)+C cdot x cdot(x-1)`

Putting `x=1`
`Rightarrow 5-1=A times 0+B(1)(1+1)+ `

`Rightarrow 4=B(2) `

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