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人気のある微分積分 >

integral of 1/(x(1+x^3))

解

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

解

\frac{1}{3} (ln x^{3} - ln 1 + x^{3}) + C

解答ステップ

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

u置換積分法を適用する

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

代用を戻す

u = 1 + x^{3}

= \frac{1}{3} (ln 1 + x^{3} - 1 - ln 1 + x^{3})

簡素化

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

= \frac{1}{3} (ln x^{3} - ln 1 + x^{3})

定数を解答に追加する

= \frac{1}{3} (ln x^{3} - ln 1 + x^{3}) + C

グラフ

人気の例

面積 y=5sqrt(x),y=4,2y+x=6

a re a y = 5 x, y = 4, 2 y + x = 6

integral from 0 to 1 of xsqrt(1+x^2)

\int_{0}^{1} x 1 + x^{2} d x

derivative x^6(1-1/(x+5))

d er i v a t i v e x^{6} (1 - \frac{1}{x + 5})

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\int_{1}^{8} \frac{7 ln ( x )}{x ^{2}} d x

limit as x approaches 2 of 1/x-1/(x^2)

x \to 2 lim (\frac{1}{x} - \frac{1}{x ^{2}})