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人気のある微分積分 >

integral of (6x)/(x^3-8)

解

\int \frac{6 x}{x ^{3} - 8} d x

解

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

解答ステップ

\int \frac{6 x}{x ^{3} - 8} d x

定数を除く:

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

= 6 \cdot \int \frac{x}{x ^{3} - 8} d x

以下の部分分数を得る:

\frac{x}{x ^{3} - 8} : \frac{1}{6 ( x - 2 )} + \frac{- x + 2}{6 ( x ^{2} + 2 x + 4 )}

= 6 \cdot \int \frac{1}{6 ( x - 2 )} + \frac{- x + 2}{6 ( x ^{2} + 2 x + 4 )} 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

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

\int \frac{1}{6 ( x - 2 )} d x = \frac{1}{6} ln ∣ x - 2 ∣

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

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

簡素化

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

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

定数を解答に追加する

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

グラフ

人気の例

integral from 0 to 2 of (sqrt(4-x^2))

\int_{0}^{2} (4 - x^{2}) d x

laplacetransform 10x

l a pl a ce t r an s f or m 10 x

(dy)/(dx)+2xy=2xe^{-x^2}

\frac{d y}{d x} + 2 x y = 2 x e^{- x^{2}}

integral of cos(y^2)

\int cos (y^{2}) d y

8y^{''}+32y^'=0,y(2)=1,y^'(-1)=1

8 y^{^{''}} + 32 y^{^{'}} = 0, y (2) = 1, y^{^{'}} (- 1) = 1