integral of (37)/(x^3-125)

解答

\int \frac{37}{x ^{3} - 125} d x

37 (\frac{1}{75} ln ∣ x - 5 ∣ - \frac{1}{150} ln 4 x^{2} + 20 x + 100 - \frac{1}{25 3} arctan (\frac{2}{5 3} x + \frac{1}{3})) + C

求解步骤

\int \frac{37}{x ^{3} - 125} d x

提出常数:

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

= 37 \cdot \int \frac{1}{x ^{3} - 125} d x

将

\frac{1}{x ^{3} - 125}

用部份分式展开:

\frac{1}{75 ( x - 5 )} + \frac{- x - 10}{75 ( x ^{2} + 5 x + 25 )}

= 37 \cdot \int \frac{1}{75 ( x - 5 )} + \frac{- x - 10}{75 ( x ^{2} + 5 x + 25 )} 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

= 37 (\int \frac{1}{75 ( x - 5 )} d x + \int \frac{- x - 10}{75 ( x ^{2} + 5 x + 25 )} d x)

\int \frac{1}{75 ( x - 5 )} d x = \frac{1}{75} ln ∣ x - 5 ∣

\int \frac{- x - 10}{75 ( x ^{2} + 5 x + 25 )} d x = \frac{1}{150 3} (- 3 ln 4 x^{2} + 20 x + 100 - 6 arctan (\frac{2}{5 3} x + \frac{1}{3}))

= 37 (\frac{1}{75} ln ∣ x - 5 ∣ + \frac{1}{150 3} (- 3 ln 4 x^{2} + 20 x + 100 - 6 arctan (\frac{2}{5 3} x + \frac{1}{3})))

乘开

\frac{1}{150 3} (- 3 ln 4 x^{2} + 20 x + 100 - 6 arctan (\frac{2}{5 3} x + \frac{1}{3})) : - \frac{1}{150} ln 4 x^{2} + 20 x + 100 - \frac{1}{25 3} arctan (\frac{2}{5 3} x + \frac{1}{3})

= 37 (\frac{1}{75} ln ∣ x - 5 ∣ - \frac{1}{150} ln 4 x^{2} + 20 x + 100 - \frac{1}{25 3} arctan (\frac{2}{5 3} x + \frac{1}{3}))

解答补常数

= 37 (\frac{1}{75} ln ∣ x - 5 ∣ - \frac{1}{150} ln 4 x^{2} + 20 x + 100 - \frac{1}{25 3} arctan (\frac{2}{5 3} x + \frac{1}{3})) + C

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