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Question Number 172712 by ilhamQ last updated on 30/Jun/22

$$1.\int \frac{x}{x^2-x-6}dx=...$$

Answered by Ar Brandon last updated on 30/Jun/22

$$=\frac{1}{2}\int \frac{2x-1}{x^2-x-6}dx+\frac{1}{2}\int \frac{1}{x^2-x-6}dx$$$$=\frac{1}{2}\int \frac{2x-1}{x^2-x-6}+\frac{1}{2}\int \frac{1}{{(x-\frac{1}{2})}^2-{\left(\frac{5}{2}\right)}^2}dx$$$$=\ln \sqrt{x^2-x-6}+\frac{1}{10}\ln \mid \frac{2x-6}{2x+4}\mid +C$$

Answered by thfchristopher last updated on 30/Jun/22

$$\frac{x}{x^2-x-6}$$$$=\frac{x}{(x-3)(x+2)}=\frac{A}{x-3}+\frac{B}{x+2}$$$$\mathrm{By}\mathrm{solving},A=\frac{3}{5},B=\frac{2}{5}$$$$\therefore \int \frac{x}{x^2-x-6}dx$$$$=\frac{1}{5}\int (\frac{3}{x-3}+\frac{2}{x-2})dx$$$$=\frac{3}{5}\int \frac{dx}{x-3}+\frac{2}{5}\int \frac{dx}{x+2}$$$$=\frac{3}{5}\ln \mid x-3\mid +\frac{2}{5}\ln \mid x+2\mid +C$$

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