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人気のある微分積分 >

integral of 5/(x^3-343)

解

\int \frac{5}{x ^{3} - 343} d x

解

5 (\frac{1}{147} ln ∣ x - 7 ∣ - \frac{1}{294} ln 4 x^{2} + 28 x + 196 - \frac{1}{49 3} arctan (\frac{2}{7 3} x + \frac{1}{3})) + C

解答ステップ

\int \frac{5}{x ^{3} - 343} d x

定数を除く:

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

= 5 \cdot \int \frac{1}{x ^{3} - 343} d x

以下の部分分数を得る:

\frac{1}{x ^{3} - 343} : \frac{1}{147 ( x - 7 )} + \frac{- x - 14}{147 ( x ^{2} + 7 x + 49 )}

= 5 \cdot \int \frac{1}{147 ( x - 7 )} + \frac{- x - 14}{147 ( x ^{2} + 7 x + 49 )} 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

= 5 (\int \frac{1}{147 ( x - 7 )} d x + \int \frac{- x - 14}{147 ( x ^{2} + 7 x + 49 )} d x)

\int \frac{1}{147 ( x - 7 )} d x = \frac{1}{147} ln ∣ x - 7 ∣

\int \frac{- x - 14}{147 ( x ^{2} + 7 x + 49 )} d x = \frac{1}{294 3} (- 3 ln 4 x^{2} + 28 x + 196 - 6 arctan (\frac{2}{7 3} x + \frac{1}{3}))

= 5 (\frac{1}{147} ln ∣ x - 7 ∣ + \frac{1}{294 3} (- 3 ln 4 x^{2} + 28 x + 196 - 6 arctan (\frac{2}{7 3} x + \frac{1}{3})))

拡張

\frac{1}{294 3} (- 3 ln 4 x^{2} + 28 x + 196 - 6 arctan (\frac{2}{7 3} x + \frac{1}{3})) : - \frac{1}{294} ln 4 x^{2} + 28 x + 196 - \frac{1}{49 3} arctan (\frac{2}{7 3} x + \frac{1}{3})

= 5 (\frac{1}{147} ln ∣ x - 7 ∣ - \frac{1}{294} ln 4 x^{2} + 28 x + 196 - \frac{1}{49 3} arctan (\frac{2}{7 3} x + \frac{1}{3}))

定数を解答に追加する

= 5 (\frac{1}{147} ln ∣ x - 7 ∣ - \frac{1}{294} ln 4 x^{2} + 28 x + 196 - \frac{1}{49 3} arctan (\frac{2}{7 3} x + \frac{1}{3})) + C

グラフ

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