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人気のある微分積分 >

integral from 1 to 4 of x/(3+sqrt(x))

解

\int_{1}^{4} \frac{x}{3 + x} d x

解

2 (- 27 (ln (5) - 2 ln (2)) + \frac{41}{6})

+1

十進法表記

1.61691 \dots

解答ステップ

\int_{1}^{4} \frac{x}{3 + x} d x

u置換積分法を適用する

= \int_{1}^{2} \frac{2 u ^{3}}{3 + u} d u

定数を除く:

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

= 2 \cdot \int_{1}^{2} \frac{u ^{3}}{3 + u} d u

長除法

\frac{u ^{3}}{3 + u} : u^{2} - 3 u + 9 - \frac{27}{u + 3}

= 2 \cdot \int_{1}^{2} u^{2} - 3 u + 9 - \frac{27}{u + 3} 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

= 2 (\int_{1}^{2} u^{2} d u - \int_{1}^{2} 3 u d u + \int_{1}^{2} 9 d u - \int_{1}^{2} \frac{27}{u + 3} d u)

\int_{1}^{2} u^{2} d u = \frac{7}{3}

\int_{1}^{2} 3 u d u = \frac{9}{2}

\int_{1}^{2} 9 d u = 9

\int_{1}^{2} \frac{27}{u + 3} d u = 27 (ln (5) - 2 ln (2))

= 2 (\frac{7}{3} - \frac{9}{2} + 9 - 27 (ln (5) - 2 ln (2)))

分数を組み合わせる

9 + \frac{7}{3} - \frac{9}{2} : \frac{41}{6}

= 2 (- 27 (ln (5) - 2 ln (2)) + \frac{41}{6})

グラフ

人気の例

integral from-1 to 1 of (x-y)

\int_{- 1}^{1} (x - y) d y

integral from 1 to 2 of (x+2)/8

\int_{1}^{2} \frac{x + 2}{8} d x

integral from-3 to 3 of (2+sqrt(9-x^2))

\int_{- 3}^{3} (2 + 9 - x^{2}) d x

integral from 0 to 6 of 2piy(36-y^2)

\int_{0}^{6} 2 π y (36 - y^{2}) d y

integral from 2 to 4 of 4/x

\int_{2}^{4} \frac{4}{x} d x