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人気のある微分積分 >

integral from 0 to 8 of 1/(v^{1/3)+1}

解

\int_{0}^{8} \frac{1}{v ^{\frac{1}{3}} + 1} d v

解

3 ln (3)

+1

十進法表記

3.29583 \dots

解答ステップ

\int_{0}^{8} \frac{1}{v ^{\frac{1}{3}} + 1} d v

u置換積分法を適用する

= \int_{0}^{2} \frac{3 u ^{2}}{u + 1} d u

定数を除く:

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

= 3 \cdot \int_{0}^{2} \frac{u ^{2}}{u + 1} d u

u置換積分法を適用する

= 3 \cdot \int_{1}^{3} \frac{( w - 1 ) ^{2}}{w} d w

拡張

\frac{( w - 1 ) ^{2}}{w} : w - 2 + \frac{1}{w}

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

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

= 3 (\int_{1}^{3} w d w - \int_{1}^{3} 2 d w + \int_{1}^{3} \frac{1}{w} d w)

\int_{1}^{3} w d w = 4

\int_{1}^{3} 2 d w = 4

\int_{1}^{3} \frac{1}{w} d w = ln (3)

= 3 (4 - 4 + ln (3))

簡素化

= 3 ln (3)

グラフ

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