integral of (3x^3-18x+15)/(x^2-x-6)

解答

\int \frac{3 x ^{3} - 18 x + 15}{x ^{2} - x - 6} d x

\frac{3 x ^{2}}{2} + 3 x + \frac{3}{2} ln 4 x^{2} - 4 x - 24 - \frac{33}{5} ln ∣ x + 2 ∣ + \frac{33}{5} ln ∣ x - 3 ∣ - \frac{21}{10} (ln \frac{2}{5} x + \frac{4}{5} - ln \frac{2}{5} x - \frac{6}{5}) + \frac{9}{5} (ln \frac{2}{5} x + \frac{4}{5} - ln \frac{2}{5} x - \frac{6}{5}) + C

求解步骤

\int \frac{3 x ^{3} - 18 x + 15}{x ^{2} - x - 6} d x

乘开

\frac{3 x ^{3} - 18 x + 15}{x ^{2} - x - 6} : \frac{3 x ^{3}}{x ^{2} - x - 6} - \frac{18 x}{x ^{2} - x - 6} + \frac{15}{x ^{2} - x - 6}

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{3 x ^{3}}{x ^{2} - x - 6} d x - \int \frac{18 x}{x ^{2} - x - 6} d x + \int \frac{15}{x ^{2} - x - 6} d x

\int \frac{3 x ^{3}}{x ^{2} - x - 6} d x = 3 (\frac{x ^{2}}{2} + x + 14 (\frac{1}{4} ln 4 x^{2} - 4 x - 24 - \frac{1}{20} (ln \frac{4}{5} + \frac{2}{5} x - ln - \frac{6}{5} + \frac{2}{5} x)) + 6 (- \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣))

\int \frac{18 x}{x ^{2} - x - 6} d x = 36 (\frac{1}{4} ln 4 x^{2} - 4 x - 24 - \frac{1}{20} (ln \frac{4}{5} + \frac{2}{5} x - ln - \frac{6}{5} + \frac{2}{5} x))

\int \frac{15}{x ^{2} - x - 6} d x = 15 (- \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣)

= 3 (\frac{x ^{2}}{2} + x + 14 (\frac{1}{4} ln 4 x^{2} - 4 x - 24 - \frac{1}{20} (ln \frac{4}{5} + \frac{2}{5} x - ln - \frac{6}{5} + \frac{2}{5} x)) + 6 (- \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣)) - 36 (\frac{1}{4} ln 4 x^{2} - 4 x - 24 - \frac{1}{20} (ln \frac{4}{5} + \frac{2}{5} x - ln - \frac{6}{5} + \frac{2}{5} x)) + 15 (- \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣)

化简

3 (\frac{x ^{2}}{2} + x + 14 (\frac{1}{4} ln 4 x^{2} - 4 x - 24 - \frac{1}{20} (ln \frac{4}{5} + \frac{2}{5} x - ln - \frac{6}{5} + \frac{2}{5} x)) + 6 (- \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣)) - 36 (\frac{1}{4} ln 4 x^{2} - 4 x - 24 - \frac{1}{20} (ln \frac{4}{5} + \frac{2}{5} x - ln - \frac{6}{5} + \frac{2}{5} x)) + 15 (- \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣) : \frac{3 x ^{2}}{2} + 3 x + \frac{3}{2} ln 4 x^{2} - 4 x - 24 - \frac{33}{5} ln ∣ x + 2 ∣ + \frac{33}{5} ln ∣ x - 3 ∣ - \frac{21}{10} (ln \frac{2}{5} x + \frac{4}{5} - ln \frac{2}{5} x - \frac{6}{5}) + \frac{9}{5} (ln \frac{2}{5} x + \frac{4}{5} - ln \frac{2}{5} x - \frac{6}{5})

= \frac{3 x ^{2}}{2} + 3 x + \frac{3}{2} ln 4 x^{2} - 4 x - 24 - \frac{33}{5} ln ∣ x + 2 ∣ + \frac{33}{5} ln ∣ x - 3 ∣ - \frac{21}{10} (ln \frac{2}{5} x + \frac{4}{5} - ln \frac{2}{5} x - \frac{6}{5}) + \frac{9}{5} (ln \frac{2}{5} x + \frac{4}{5} - ln \frac{2}{5} x - \frac{6}{5})

解答补常数

= \frac{3 x ^{2}}{2} + 3 x + \frac{3}{2} ln 4 x^{2} - 4 x - 24 - \frac{33}{5} ln ∣ x + 2 ∣ + \frac{33}{5} ln ∣ x - 3 ∣ - \frac{21}{10} (ln \frac{2}{5} x + \frac{4}{5} - ln \frac{2}{5} x - \frac{6}{5}) + \frac{9}{5} (ln \frac{2}{5} x + \frac{4}{5} - ln \frac{2}{5} x - \frac{6}{5}) + C

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