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

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

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`=int frac{x^{3}}{(x-1)(x^{2}-5 x+6)} d x`
`=int frac{x^{3}}{x^{3}-5 x^{2}+6 x-x^{2}+5 x-6} d x`
`=int frac{x^{3}}{x^{3}-6 x^{2}+11 x-6} d x`
`therefore frac{x^{3}}{x^{3}-6 x^{2}+11 x-6}=1+frac{6 x^{2}+11 x+6}{x^{2}-6 x^{2}+11 x-6}`
`Rightarrow frac{6 x^{2}-11 x+6}{x^{3}-6 x^{2}+11 x-6}=1+frac{110}{(x-1)(x-2)(x-3)}`
`therefore int frac{x^{3}}{(x-1)(x-2)(x-3)} d x=int d x+int frac{(6 x^{2}-11 x+6)}{(x-1)(x-2)(x-3)} d x ldots ldots ldots ldots` (1)
Let `frac{6 x^{2}-11 x+6}{(x-1)(x-2)(x-3)}=frac{A}{x-1}+frac{B}{x-2}+frac{C}{x-3}`
`Rightarrow frac{6 x^{2}-11 x+6}{(x-1)(x-2)(x-3)}=frac{A(x-2)(x-3)+B(x-1)(x-3)+C(x-1)(x-2)}{(x-1)(x-2)(x-3)}`
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