integral of (x^2+2x-1)/(x^3-27)

解答

\int \frac{x ^{2} + 2 x - 1}{x ^{3} - 27} d x

\frac{1}{3} ln x^{3} - 27 + \frac{5}{27} ln ∣ x - 3 ∣ + \frac{1}{54} ln 4 x^{2} + 12 x + 36 + \frac{1}{9 3} arctan (\frac{2}{3 3} x + \frac{1}{3}) + \frac{1}{9} (- ln 4 x^{2} + 12 x + 36 + 23 arctan (\frac{2}{3 3} x + \frac{1}{3})) + C

求解步骤

\int \frac{x ^{2} + 2 x - 1}{x ^{3} - 27} d x

乘开

\frac{x ^{2} + 2 x - 1}{x ^{3} - 27} : \frac{x ^{2}}{x ^{3} - 27} + \frac{2 x}{x ^{3} - 27} - \frac{1}{x ^{3} - 27}

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{x ^{2}}{x ^{3} - 27} d x + \int \frac{2 x}{x ^{3} - 27} d x - \int \frac{1}{x ^{3} - 27} d x

\int \frac{x ^{2}}{x ^{3} - 27} d x = \frac{1}{3} ln x^{3} - 27

\int \frac{2 x}{x ^{3} - 27} d x = 2 (\frac{1}{9} ln ∣ x - 3 ∣ + \frac{1}{18} (- ln 4 x^{2} + 12 x + 36 + 23 arctan (\frac{1}{3 3} (2 x + 3))))

\int \frac{1}{x ^{3} - 27} d x = \frac{1}{27} ln ∣ x - 3 ∣ - \frac{1}{54} ln 4 x^{2} + 12 x + 36 - \frac{1}{9 3} arctan (\frac{2}{3 3} x + \frac{1}{3})

= \frac{1}{3} ln x^{3} - 27 + 2 (\frac{1}{9} ln ∣ x - 3 ∣ + \frac{1}{18} (- ln 4 x^{2} + 12 x + 36 + 23 arctan (\frac{1}{3 3} (2 x + 3)))) - (\frac{1}{27} ln ∣ x - 3 ∣ - \frac{1}{54} ln 4 x^{2} + 12 x + 36 - \frac{1}{9 3} arctan (\frac{2}{3 3} x + \frac{1}{3}))

化简

\frac{1}{3} ln x^{3} - 27 + 2 (\frac{1}{9} ln ∣ x - 3 ∣ + \frac{1}{18} (- ln 4 x^{2} + 12 x + 36 + 23 arctan (\frac{1}{3 3} (2 x + 3)))) - (\frac{1}{27} ln ∣ x - 3 ∣ - \frac{1}{54} ln 4 x^{2} + 12 x + 36 - \frac{1}{9 3} arctan (\frac{2}{3 3} x + \frac{1}{3})) : \frac{1}{3} ln x^{3} - 27 + \frac{5}{27} ln ∣ x - 3 ∣ + \frac{1}{54} ln 4 x^{2} + 12 x + 36 + \frac{1}{9 3} arctan (\frac{2}{3 3} x + \frac{1}{3}) + \frac{1}{9} (- ln 4 x^{2} + 12 x + 36 + 23 arctan (\frac{2}{3 3} x + \frac{1}{3}))

= \frac{1}{3} ln x^{3} - 27 + \frac{5}{27} ln ∣ x - 3 ∣ + \frac{1}{54} ln 4 x^{2} + 12 x + 36 + \frac{1}{9 3} arctan (\frac{2}{3 3} x + \frac{1}{3}) + \frac{1}{9} (- ln 4 x^{2} + 12 x + 36 + 23 arctan (\frac{2}{3 3} x + \frac{1}{3}))

解答补常数

= \frac{1}{3} ln x^{3} - 27 + \frac{5}{27} ln ∣ x - 3 ∣ + \frac{1}{54} ln 4 x^{2} + 12 x + 36 + \frac{1}{9 3} arctan (\frac{2}{3 3} x + \frac{1}{3}) + \frac{1}{9} (- ln 4 x^{2} + 12 x + 36 + 23 arctan (\frac{2}{3 3} x + \frac{1}{3})) + C

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