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人気のある微分積分 >

integral of \sqrt[7]{x^3}(1+sqrt(x))

解

解

\frac{1}{3} (\frac{21}{10} x^{\frac{10}{7}} + \frac{14}{9} x^{\frac{27}{14}}) + C

解答ステップ

u置換積分法を適用する

定数を除く:

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

拡張

= \frac{1}{3} \cdot \int \frac{1}{u ^{\frac{11}{21}}} + \frac{1}{u ^{\frac{5}{14}}} 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 ^{\frac{11}{21}}} d u + \int \frac{1}{u ^{\frac{5}{14}}} d u)

\int \frac{1}{u ^{\frac{11}{21}}} d u = \frac{21}{10} u^{\frac{10}{21}}

\int \frac{1}{u ^{\frac{5}{14}}} d u = \frac{14}{9} u^{\frac{9}{14}}

= \frac{1}{3} (\frac{21}{10} u^{\frac{10}{21}} + \frac{14}{9} u^{\frac{9}{14}})

代用を戻す

u = x^{3}

= \frac{1}{3} (\frac{21}{10} (x^{3})^{\frac{10}{21}} + \frac{14}{9} (x^{3})^{\frac{9}{14}})

簡素化

\frac{1}{3} (\frac{21}{10} (x^{3})^{\frac{10}{21}} + \frac{14}{9} (x^{3})^{\frac{9}{14}}) : \frac{1}{3} (\frac{21}{10} x^{\frac{10}{7}} + \frac{14}{9} x^{\frac{27}{14}})

= \frac{1}{3} (\frac{21}{10} x^{\frac{10}{7}} + \frac{14}{9} x^{\frac{27}{14}})

定数を解答に追加する

= \frac{1}{3} (\frac{21}{10} x^{\frac{10}{7}} + \frac{14}{9} x^{\frac{27}{14}}) + C

グラフ

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