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人気のある微分積分 >

integral of (17)/(x^3-125)

解

\int \frac{17}{x ^{3} - 125} d x

解

17 (\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

解答ステップ

\int \frac{17}{x ^{3} - 125} d x

定数を除く:

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

= 17 \cdot \int \frac{1}{x ^{3} - 125} d x

以下の部分分数を得る:

\frac{1}{x ^{3} - 125} : \frac{1}{75 ( x - 5 )} + \frac{- x - 10}{75 ( x ^{2} + 5 x + 25 )}

= 17 \cdot \int \frac{1}{75 ( x - 5 )} + \frac{- x - 10}{75 ( x ^{2} + 5 x + 25 )} d x

総和規則を適用する:

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

= 17 (\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}))

= 17 (\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})))

拡張

\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})

= 17 (\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}))

定数を解答に追加する

= 17 (\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

グラフ

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