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人気のある微分積分 >

integral of 3/(x^3-1)

解

\int \frac{3}{x ^{3} - 1} d x

解

ln ∣ x - 1 ∣ - \frac{1}{2} ln 4 x^{2} + 4 x + 4 - 3 arctan (\frac{1}{3} (2 x + 1)) + C

解答ステップ

\int \frac{3}{x ^{3} - 1} d x

定数を除く:

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

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

以下の部分分数を得る:

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

= 3 \cdot \int \frac{1}{3 ( x - 1 )} + \frac{- x - 2}{3 ( x ^{2} + x + 1 )} 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

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

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

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

= 3 (\frac{1}{3} ln ∣ x - 1 ∣ + \frac{1}{6 3} (- 3 ln 4 x^{2} + 4 x + 4 - 6 arctan (\frac{2 x + 1}{3})))

簡素化

3 (\frac{1}{3} ln ∣ x - 1 ∣ + \frac{1}{6 3} (- 3 ln 4 x^{2} + 4 x + 4 - 6 arctan (\frac{2 x + 1}{3}))) : ln ∣ x - 1 ∣ - \frac{1}{2} ln 4 x^{2} + 4 x + 4 - 3 arctan (\frac{1}{3} (2 x + 1))

= ln ∣ x - 1 ∣ - \frac{1}{2} ln 4 x^{2} + 4 x + 4 - 3 arctan (\frac{1}{3} (2 x + 1))

定数を解答に追加する

= ln ∣ x - 1 ∣ - \frac{1}{2} ln 4 x^{2} + 4 x + 4 - 3 arctan (\frac{1}{3} (2 x + 1)) + C

グラフ

人気の例

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\int sin^{3} (9 x) cos (9 x) d x

integral of sqrt(R^2-x^2)

\int R^{2} - x^{2} d x

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\int \frac{x ^{2}}{x ^{2} - 16} d x

integral of (x+1)sqrt(x)

\int (x + 1) x d x

integral of 1/(4-9t^2)

\int \frac{1}{4 - 9 t ^{2}} d t