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人気のある微分積分 >

integral of ((4+x^7)^67x^6)

解

\int ((4 + x^{7})^{6} 7 x^{6}) d x

解

4096 x^{7} + 3072 x^{14} + 1280 x^{21} + 320 x^{28} + 48 x^{35} + 4 x^{42} + \frac{x ^{49}}{7} + C

解答ステップ

\int (4 + x^{7})^{6} \cdot 7 x^{6} d x

定数を除く:

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

= 7 \cdot \int (4 + x^{7})^{6} x^{6} d x

拡張

(4 + x^{7})^{6} x^{6} : 4096 x^{6} + 6144 x^{13} + 3840 x^{20} + 1280 x^{27} + 240 x^{34} + 24 x^{41} + x^{48}

= 7 \cdot \int 4096 x^{6} + 6144 x^{13} + 3840 x^{20} + 1280 x^{27} + 240 x^{34} + 24 x^{41} + x^{48} d x

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

= 7 (\int 4096 x^{6} d x + \int 6144 x^{13} d x + \int 3840 x^{20} d x + \int 1280 x^{27} d x + \int 240 x^{34} d x + \int 24 x^{41} d x + \int x^{48} d x)

\int 4096 x^{6} d x = \frac{4096 x ^{7}}{7}

\int 6144 x^{13} d x = \frac{3072 x ^{14}}{7}

\int 3840 x^{20} d x = \frac{1280 x ^{21}}{7}

\int 1280 x^{27} d x = \frac{320 x ^{28}}{7}

\int 240 x^{34} d x = \frac{48 x ^{35}}{7}

\int 24 x^{41} d x = \frac{4 x ^{42}}{7}

\int x^{48} d x = \frac{x ^{49}}{49}

= 7 (\frac{4096 x ^{7}}{7} + \frac{3072 x ^{14}}{7} + \frac{1280 x ^{21}}{7} + \frac{320 x ^{28}}{7} + \frac{48 x ^{35}}{7} + \frac{4 x ^{42}}{7} + \frac{x ^{49}}{49})

簡素化

7 (\frac{4096 x ^{7}}{7} + \frac{3072 x ^{14}}{7} + \frac{1280 x ^{21}}{7} + \frac{320 x ^{28}}{7} + \frac{48 x ^{35}}{7} + \frac{4 x ^{42}}{7} + \frac{x ^{49}}{49}) : 4096 x^{7} + 3072 x^{14} + 1280 x^{21} + 320 x^{28} + 48 x^{35} + 4 x^{42} + \frac{x ^{49}}{7}

= 4096 x^{7} + 3072 x^{14} + 1280 x^{21} + 320 x^{28} + 48 x^{35} + 4 x^{42} + \frac{x ^{49}}{7}

定数を解答に追加する

= 4096 x^{7} + 3072 x^{14} + 1280 x^{21} + 320 x^{28} + 48 x^{35} + 4 x^{42} + \frac{x ^{49}}{7} + C

グラフ

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