` int[3x+5]/[x^3-x^2-x+1].dx`

Explanation

`I=int frac{(3 x+5) d x}{x^{3}-x^{2}-x+1} `

`=int frac{(3 x+5) d x}{x^{2}(x-1)-1(x-1)} `

`=int frac{(3 x+5) d x}{(x^{2}-1)(x-1)} `

`=int frac{(3 x+5) d x}{(x-1)(x+1)(x-1)} `

`=int frac{(3 x+5) d x}{(x-1)^{2}(x+1)} `

`text { Let } frac{3 x+5}{(x-1)^{2}(x+1)}=frac{A}{x+1}+frac{B}{x-1}+frac{C}{(x-1)^{2}} `

`Rightarrow frac{3 x+5}{(x-1)^{2}(x+1)}=frac{A(x-1)^{2}+B(x+1)(x-1)+C(x+1)}{(x+1)(x-1)^{2}} `

`Rightarrow 3 x+5=A(x^{2}-2 x+1)+B(x^{2}-1)+C x+C `

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