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人気のある微分積分 >

integral of 7/(x(x^4+6))

解

\int \frac{7}{x ( x ^{4} + 6 )} d x

解

\frac{7}{4} (\frac{2}{3} ln (x) - \frac{1}{6} ln x^{4} + 6) + C

解答ステップ

\int \frac{7}{x ( x ^{4} + 6 )} d x

定数を除く:

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

= 7 \cdot \int \frac{1}{x ( x ^{4} + 6 )} d x

u置換積分法を適用する

= 7 \cdot \int \frac{1}{4 u ( u + 6 )} d u

定数を除く:

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

= 7 \cdot \frac{1}{4} \cdot \int \frac{1}{u ( u + 6 )} d u

以下の部分分数を得る:

\frac{1}{u ( u + 6 )} : \frac{1}{6 u} - \frac{1}{6 ( u + 6 )}

= 7 \cdot \frac{1}{4} \cdot \int \frac{1}{6 u} - \frac{1}{6 ( u + 6 )} 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

= 7 \cdot \frac{1}{4} (\int \frac{1}{6 u} d u - \int \frac{1}{6 ( u + 6 )} d u)

\int \frac{1}{6 u} d u = \frac{1}{6} ln ∣ u ∣

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

= 7 \cdot \frac{1}{4} (\frac{1}{6} ln ∣ u ∣ - \frac{1}{6} ln ∣ u + 6 ∣)

代用を戻す

u = x^{4}

= 7 \cdot \frac{1}{4} (\frac{1}{6} ln x^{4} - \frac{1}{6} ln x^{4} + 6)

簡素化

7 \cdot \frac{1}{4} (\frac{1}{6} ln x^{4} - \frac{1}{6} ln x^{4} + 6) : \frac{7}{4} (\frac{2}{3} ln (x) - \frac{1}{6} ln x^{4} + 6)

= \frac{7}{4} (\frac{2}{3} ln (x) - \frac{1}{6} ln x^{4} + 6)

定数を解答に追加する

= \frac{7}{4} (\frac{2}{3} ln (x) - \frac{1}{6} ln x^{4} + 6) + C

グラフ

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