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人気のある微分積分 >

integral of 6/(x(x^2+6)^2)

解

\int \frac{6}{x ( x ^{2} + 6 ) ^{2}} d x

解

3 (\frac{1}{18} ln (x) - \frac{1}{36} ln x^{2} + 6 + \frac{1}{6 ( x ^{2} + 6 )}) + C

解答ステップ

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

u置換積分法を適用する

= 6 \cdot \int \frac{1}{2 ( u - 6 ) u ^{2}} d u

定数を除く:

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

= 6 \cdot \frac{1}{2} \cdot \int \frac{1}{( u - 6 ) u ^{2}} d u

以下の部分分数を得る:

\frac{1}{( u - 6 ) u ^{2}} : \frac{1}{36 ( u - 6 )} - \frac{1}{36 u} - \frac{1}{6 u ^{2}}

= 6 \cdot \frac{1}{2} \cdot \int \frac{1}{36 ( u - 6 )} - \frac{1}{36 u} - \frac{1}{6 u ^{2}} d u

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 \cdot \frac{1}{2} (\int \frac{1}{36 ( u - 6 )} d u - \int \frac{1}{36 u} d u - \int \frac{1}{6 u ^{2}} d u)

\int \frac{1}{36 ( u - 6 )} d u = \frac{1}{36} ln ∣ u - 6 ∣

\int \frac{1}{36 u} d u = \frac{1}{36} ln ∣ u ∣

\int \frac{1}{6 u ^{2}} d u = - \frac{1}{6 u}

= 6 \cdot \frac{1}{2} (\frac{1}{36} ln ∣ u - 6 ∣ - \frac{1}{36} ln ∣ u ∣ - (- \frac{1}{6 u}))

代用を戻す

u = x^{2} + 6

= 6 \cdot \frac{1}{2} (\frac{1}{36} ln x^{2} + 6 - 6 - \frac{1}{36} ln x^{2} + 6 - (- \frac{1}{6 ( x ^{2} + 6 )}))

簡素化

6 \cdot \frac{1}{2} (\frac{1}{36} ln x^{2} + 6 - 6 - \frac{1}{36} ln x^{2} + 6 - (- \frac{1}{6 ( x ^{2} + 6 )})) : 3 (\frac{1}{18} ln (x) - \frac{1}{36} ln x^{2} + 6 + \frac{1}{6 ( x ^{2} + 6 )})

= 3 (\frac{1}{18} ln (x) - \frac{1}{36} ln x^{2} + 6 + \frac{1}{6 ( x ^{2} + 6 )})

定数を解答に追加する

= 3 (\frac{1}{18} ln (x) - \frac{1}{36} ln x^{2} + 6 + \frac{1}{6 ( x ^{2} + 6 )}) + C

グラフ

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