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人気のある微分積分 >

integral of x^5(x^6-10)^4

解

\int x^{5} (x^{6} - 10)^{4} d x

解

\frac{1}{30} (x^{30} - 50 x^{24} + 1000 x^{18} - 10000 x^{12} + 50000 x^{6}) + C

解答ステップ

\int x^{5} (x^{6} - 10)^{4} d x

u置換積分法を適用する

= \int \frac{u ^{\frac{1}{5}} ( u ^{\frac{6}{5}} - 10 ) ^{4}}{5} d u

定数を除く:

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

= \frac{1}{5} \cdot \int u^{\frac{1}{5}} (u^{\frac{6}{5}} - 10)^{4} d u

拡張

u^{\frac{1}{5}} (u^{\frac{6}{5}} - 10)^{4} : u^{5} - 40 u^{\frac{19}{5}} + 600 u^{\frac{13}{5}} - 4000 u^{\frac{7}{5}} + 10000 u^{\frac{1}{5}}

= \frac{1}{5} \cdot \int u^{5} - 40 u^{\frac{19}{5}} + 600 u^{\frac{13}{5}} - 4000 u^{\frac{7}{5}} + 10000 u^{\frac{1}{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 u^{5} d u - \int 40 u^{\frac{19}{5}} d u + \int 600 u^{\frac{13}{5}} d u - \int 4000 u^{\frac{7}{5}} d u + \int 10000 u^{\frac{1}{5}} d u)

\int u^{5} d u = \frac{u ^{6}}{6}

\int 40 u^{\frac{19}{5}} d u = \frac{25}{3} u^{\frac{24}{5}}

\int 600 u^{\frac{13}{5}} d u = \frac{500}{3} u^{\frac{18}{5}}

\int 4000 u^{\frac{7}{5}} d u = \frac{5000}{3} u^{\frac{12}{5}}

\int 10000 u^{\frac{1}{5}} d u = \frac{25000}{3} u^{\frac{6}{5}}

= \frac{1}{5} (\frac{u ^{6}}{6} - \frac{25}{3} u^{\frac{24}{5}} + \frac{500}{3} u^{\frac{18}{5}} - \frac{5000}{3} u^{\frac{12}{5}} + \frac{25000}{3} u^{\frac{6}{5}})

代用を戻す

u = x^{5}

= \frac{1}{5} (\frac{( x ^{5} ) ^{6}}{6} - \frac{25}{3} (x^{5})^{\frac{24}{5}} + \frac{500}{3} (x^{5})^{\frac{18}{5}} - \frac{5000}{3} (x^{5})^{\frac{12}{5}} + \frac{25000}{3} (x^{5})^{\frac{6}{5}})

簡素化

\frac{1}{5} (\frac{( x ^{5} ) ^{6}}{6} - \frac{25}{3} (x^{5})^{\frac{24}{5}} + \frac{500}{3} (x^{5})^{\frac{18}{5}} - \frac{5000}{3} (x^{5})^{\frac{12}{5}} + \frac{25000}{3} (x^{5})^{\frac{6}{5}}) : \frac{1}{30} (x^{30} - 50 x^{24} + 1000 x^{18} - 10000 x^{12} + 50000 x^{6})

= \frac{1}{30} (x^{30} - 50 x^{24} + 1000 x^{18} - 10000 x^{12} + 50000 x^{6})

定数を解答に追加する

= \frac{1}{30} (x^{30} - 50 x^{24} + 1000 x^{18} - 10000 x^{12} + 50000 x^{6}) + C

グラフ

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