integral of 5/(x^3-27)

解答

\int \frac{5}{x ^{3} - 27} d x

5 (\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})) + C

求解步骤

\int \frac{5}{x ^{3} - 27} d x

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 5 \cdot \int \frac{1}{x ^{3} - 27} d x

将

\frac{1}{x ^{3} - 27}

用部份分式展开:

\frac{1}{27 ( x - 3 )} + \frac{- x - 6}{27 ( x ^{2} + 3 x + 9 )}

= 5 \cdot \int \frac{1}{27 ( x - 3 )} + \frac{- x - 6}{27 ( x ^{2} + 3 x + 9 )} d x

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

= 5 (\int \frac{1}{27 ( x - 3 )} d x + \int \frac{- x - 6}{27 ( x ^{2} + 3 x + 9 )} d x)

\int \frac{1}{27 ( x - 3 )} d x = \frac{1}{27} ln ∣ x - 3 ∣

\int \frac{- x - 6}{27 ( x ^{2} + 3 x + 9 )} d x = \frac{1}{54 3} (- 3 ln 4 x^{2} + 12 x + 36 - 6 arctan (\frac{2}{3 3} x + \frac{1}{3}))

= 5 (\frac{1}{27} ln ∣ x - 3 ∣ + \frac{1}{54 3} (- 3 ln 4 x^{2} + 12 x + 36 - 6 arctan (\frac{2}{3 3} x + \frac{1}{3})))

乘开

\frac{1}{54 3} (- 3 ln 4 x^{2} + 12 x + 36 - 6 arctan (\frac{2}{3 3} x + \frac{1}{3})) : - \frac{1}{54} ln 4 x^{2} + 12 x + 36 - \frac{1}{9 3} arctan (\frac{2}{3 3} x + \frac{1}{3})

= 5 (\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}))

解答补常数

= 5 (\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})) + C

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