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人気のある微分積分 >

integral of (4x^2)/(\sqrt[3]{3+5x)}

解

\int \frac{4 x ^{2}}{3 3 + 5 x} d x

解

\frac{4}{125} (\frac{3}{8} (3 + 5 x)^{\frac{8}{3}} - \frac{18}{5} (3 + 5 x)^{\frac{5}{3}} + \frac{27}{2} (3 + 5 x)^{\frac{2}{3}}) + C

解答ステップ

\int \frac{4 x ^{2}}{3 3 + 5 x} d x

定数を除く:

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

= 4 \cdot \int \frac{x ^{2}}{3 3 + 5 x} d x

u置換積分法を適用する

= 4 \cdot \int \frac{( u - 3 ) ^{2}}{125 3 u} d u

定数を除く:

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

= 4 \cdot \frac{1}{125} \cdot \int \frac{( u - 3 ) ^{2}}{3 u} d u

拡張

\frac{( u - 3 ) ^{2}}{3 u} : u^{\frac{5}{3}} - 6 u^{\frac{2}{3}} + \frac{9}{3 u}

= 4 \cdot \frac{1}{125} \cdot \int u^{\frac{5}{3}} - 6 u^{\frac{2}{3}} + \frac{9}{3 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

= 4 \cdot \frac{1}{125} (\int u^{\frac{5}{3}} d u - \int 6 u^{\frac{2}{3}} d u + \int \frac{9}{3 u} d u)

\int u^{\frac{5}{3}} d u = \frac{3}{8} u^{\frac{8}{3}}

\int 6 u^{\frac{2}{3}} d u = \frac{18}{5} u^{\frac{5}{3}}

\int \frac{9}{3 u} d u = \frac{27}{2} u^{\frac{2}{3}}

= 4 \cdot \frac{1}{125} (\frac{3}{8} u^{\frac{8}{3}} - \frac{18}{5} u^{\frac{5}{3}} + \frac{27}{2} u^{\frac{2}{3}})

代用を戻す

u = 3 + 5 x

= 4 \cdot \frac{1}{125} (\frac{3}{8} (3 + 5 x)^{\frac{8}{3}} - \frac{18}{5} (3 + 5 x)^{\frac{5}{3}} + \frac{27}{2} (3 + 5 x)^{\frac{2}{3}})

簡素化

4 \cdot \frac{1}{125} (\frac{3}{8} (3 + 5 x)^{\frac{8}{3}} - \frac{18}{5} (3 + 5 x)^{\frac{5}{3}} + \frac{27}{2} (3 + 5 x)^{\frac{2}{3}}) : \frac{4}{125} (\frac{3}{8} (3 + 5 x)^{\frac{8}{3}} - \frac{18}{5} (3 + 5 x)^{\frac{5}{3}} + \frac{27}{2} (3 + 5 x)^{\frac{2}{3}})

= \frac{4}{125} (\frac{3}{8} (3 + 5 x)^{\frac{8}{3}} - \frac{18}{5} (3 + 5 x)^{\frac{5}{3}} + \frac{27}{2} (3 + 5 x)^{\frac{2}{3}})

定数を解答に追加する

= \frac{4}{125} (\frac{3}{8} (3 + 5 x)^{\frac{8}{3}} - \frac{18}{5} (3 + 5 x)^{\frac{5}{3}} + \frac{27}{2} (3 + 5 x)^{\frac{2}{3}}) + C

グラフ

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