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人気のある微分積分 >

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

解

解

\frac{1}{132} (36 x^{\frac{11}{3}} + 99 x^{\frac{8}{3}} - 44 x^{3} - 132 x^{2}) + 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}{5} \cdot \int \frac{1}{u ^{\frac{4}{15}}} + \frac{2}{u ^{\frac{7}{15}}} - \frac{1}{u ^{\frac{2}{5}}} - \frac{2}{u ^{\frac{3}{5}}} 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}{5} (\int \frac{1}{u ^{\frac{4}{15}}} d u + \int \frac{2}{u ^{\frac{7}{15}}} d u - \int \frac{1}{u ^{\frac{2}{5}}} d u - \int \frac{2}{u ^{\frac{3}{5}}} d u)

\int \frac{1}{u ^{\frac{4}{15}}} d u = \frac{15}{11} u^{\frac{11}{15}}

\int \frac{2}{u ^{\frac{7}{15}}} d u = \frac{15}{4} u^{\frac{8}{15}}

\int \frac{1}{u ^{\frac{2}{5}}} d u = \frac{5}{3} u^{\frac{3}{5}}

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

= \frac{1}{5} (\frac{15}{11} u^{\frac{11}{15}} + \frac{15}{4} u^{\frac{8}{15}} - \frac{5}{3} u^{\frac{3}{5}} - 5 u^{\frac{2}{5}})

代用を戻す

u = x^{5}

= \frac{1}{5} (\frac{15}{11} (x^{5})^{\frac{11}{15}} + \frac{15}{4} (x^{5})^{\frac{8}{15}} - \frac{5}{3} (x^{5})^{\frac{3}{5}} - 5 (x^{5})^{\frac{2}{5}})

簡素化

\frac{1}{5} (\frac{15}{11} (x^{5})^{\frac{11}{15}} + \frac{15}{4} (x^{5})^{\frac{8}{15}} - \frac{5}{3} (x^{5})^{\frac{3}{5}} - 5 (x^{5})^{\frac{2}{5}}) : \frac{1}{132} (36 x^{\frac{11}{3}} + 99 x^{\frac{8}{3}} - 44 x^{3} - 132 x^{2})

= \frac{1}{132} (36 x^{\frac{11}{3}} + 99 x^{\frac{8}{3}} - 44 x^{3} - 132 x^{2})

定数を解答に追加する

= \frac{1}{132} (36 x^{\frac{11}{3}} + 99 x^{\frac{8}{3}} - 44 x^{3} - 132 x^{2}) + C

グラフ

人気の例

derivative of ln(2-5x)

derivative of arctan(x-x^2)

integral of sin(7x)ln(sin(7x))