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Question Number 128888 by bemath last updated on 11/Jan/21

$$\int \frac{(\mathrm{x}-1)(\mathrm{x}-2)(\mathrm{x}-3)}{(\mathrm{x}-4)(\mathrm{x}-5)(\mathrm{x}-6)}\mathrm{dx}=?$$

Answered by Olaf last updated on 11/Jan/21

$$\mathrm{\Omega }=\int \frac{(x-1)(x-2)(x-3)}{(x-4)(x-5)(x-6)}dx$$$$\mathrm{\Omega }=\int (1+\frac{\mathrm{A}}{x-4}+\frac{\mathrm{B}}{x-5}+\frac{\mathrm{C}}{x-6})dx$$$$\mathrm{A}=\frac{3\times 2\times 1}{(-1)(-2)}=3$$$$\mathrm{B}=\frac{4\times 3\times 2}{1\times (-1)}=-24$$$$\mathrm{C}=\frac{5\times 4\times 3}{2\times 1}=30$$$$\mathrm{\Omega }=\int (1+\frac{3}{x-4}-\frac{24}{x-5}+\frac{30}{x-6})dx$$$$\mathrm{\Omega }=x+3\ln \mid x-4\mid -24\ln \mid x-5\mid +30\ln \mid x-6\mid +\mathrm{C}$$

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