integral of x^2(5x+2x^3)^8

解

\int x^{2} (5 x + 2 x^{3})^{8} d x

\frac{390625 x ^{11}}{11} + \frac{1250000 x ^{13}}{13} + \frac{350000 x ^{15}}{3} + \frac{1400000 x ^{17}}{17} + \frac{700000 x ^{19}}{19} + \frac{32000 x ^{21}}{3} + \frac{44800 x ^{23}}{23} + \frac{1024 x ^{25}}{5} + \frac{256 x ^{27}}{27} + C

解答ステップ

\int x^{2} (5 x + 2 x^{3})^{8} d x

拡張

x^{2} (5 x + 2 x^{3})^{8} : 390625 x^{10} + 1250000 x^{12} + 1750000 x^{14} + 1400000 x^{16} + 700000 x^{18} + 224000 x^{20} + 44800 x^{22} + 5120 x^{24} + 256 x^{26}

= \int 390625 x^{10} + 1250000 x^{12} + 1750000 x^{14} + 1400000 x^{16} + 700000 x^{18} + 224000 x^{20} + 44800 x^{22} + 5120 x^{24} + 256 x^{26} 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

= \int 390625 x^{10} d x + \int 1250000 x^{12} d x + \int 1750000 x^{14} d x + \int 1400000 x^{16} d x + \int 700000 x^{18} d x + \int 224000 x^{20} d x + \int 44800 x^{22} d x + \int 5120 x^{24} d x + \int 256 x^{26} d x

\int 390625 x^{10} d x = \frac{390625 x ^{11}}{11}

\int 1250000 x^{12} d x = \frac{1250000 x ^{13}}{13}

\int 1750000 x^{14} d x = \frac{350000 x ^{15}}{3}

\int 1400000 x^{16} d x = \frac{1400000 x ^{17}}{17}

\int 700000 x^{18} d x = \frac{700000 x ^{19}}{19}

\int 224000 x^{20} d x = \frac{32000 x ^{21}}{3}

\int 44800 x^{22} d x = \frac{44800 x ^{23}}{23}

\int 5120 x^{24} d x = \frac{1024 x ^{25}}{5}

\int 256 x^{26} d x = \frac{256 x ^{27}}{27}

= \frac{390625 x ^{11}}{11} + \frac{1250000 x ^{13}}{13} + \frac{350000 x ^{15}}{3} + \frac{1400000 x ^{17}}{17} + \frac{700000 x ^{19}}{19} + \frac{32000 x ^{21}}{3} + \frac{44800 x ^{23}}{23} + \frac{1024 x ^{25}}{5} + \frac{256 x ^{27}}{27}

定数を解答に追加する

= \frac{390625 x ^{11}}{11} + \frac{1250000 x ^{13}}{13} + \frac{350000 x ^{15}}{3} + \frac{1400000 x ^{17}}{17} + \frac{700000 x ^{19}}{19} + \frac{32000 x ^{21}}{3} + \frac{44800 x ^{23}}{23} + \frac{1024 x ^{25}}{5} + \frac{256 x ^{27}}{27} + C