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人気のある微分積分 >

integral of 1/(x(x^4+1))

解

\int \frac{1}{x ( x ^{4} + 1 )} d x

解

\frac{1}{4} (4 ln (x) - ln 1 + x^{4}) + C

解答ステップ

\int \frac{1}{x ( x ^{4} + 1 )} d x

u置換積分法を適用する

= \int \frac{1}{4 ( u - 1 ) u} d u

定数を除く:

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

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

以下の部分分数を得る:

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

= \frac{1}{4} \cdot \int \frac{1}{u - 1} - \frac{1}{u} 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

= \frac{1}{4} (\int \frac{1}{u - 1} d u - \int \frac{1}{u} d u)

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

\int \frac{1}{u} d u = ln ∣ u ∣

= \frac{1}{4} (ln ∣ u - 1 ∣ - ln ∣ u ∣)

代用を戻す

u = 1 + x^{4}

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

簡素化

\frac{1}{4} (ln 1 + x^{4} - 1 - ln 1 + x^{4}) : \frac{1}{4} (4 ln (x) - ln 1 + x^{4})

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

定数を解答に追加する

= \frac{1}{4} (4 ln (x) - ln 1 + x^{4}) + C

グラフ

人気の例

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