integral of x^4(3x^5+1)^9

解

\int x^{4} (3 x^{5} + 1)^{9} d x

\frac{19683 x ^{50}}{50} + \frac{6561 x ^{45}}{5} + \frac{19683 x ^{40}}{10} + \frac{8748 x ^{35}}{5} + \frac{5103 x ^{30}}{5} + \frac{10206 x ^{25}}{25} + \frac{567 x ^{20}}{5} + \frac{108 x ^{15}}{5} + \frac{27 x ^{10}}{10} + \frac{x ^{5}}{5} + C

解答ステップ

\int x^{4} (3 x^{5} + 1)^{9} d x

拡張

x^{4} (3 x^{5} + 1)^{9} : 19683 x^{49} + 59049 x^{44} + 78732 x^{39} + 61236 x^{34} + 30618 x^{29} + 10206 x^{24} + 2268 x^{19} + 324 x^{14} + 27 x^{9} + x^{4}

= \int 19683 x^{49} + 59049 x^{44} + 78732 x^{39} + 61236 x^{34} + 30618 x^{29} + 10206 x^{24} + 2268 x^{19} + 324 x^{14} + 27 x^{9} + x^{4} 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 19683 x^{49} d x + \int 59049 x^{44} d x + \int 78732 x^{39} d x + \int 61236 x^{34} d x + \int 30618 x^{29} d x + \int 10206 x^{24} d x + \int 2268 x^{19} d x + \int 324 x^{14} d x + \int 27 x^{9} d x + \int x^{4} d x

\int 19683 x^{49} d x = \frac{19683 x ^{50}}{50}

\int 59049 x^{44} d x = \frac{6561 x ^{45}}{5}

\int 78732 x^{39} d x = \frac{19683 x ^{40}}{10}

\int 61236 x^{34} d x = \frac{8748 x ^{35}}{5}

\int 30618 x^{29} d x = \frac{5103 x ^{30}}{5}

\int 10206 x^{24} d x = \frac{10206 x ^{25}}{25}

\int 2268 x^{19} d x = \frac{567 x ^{20}}{5}

\int 324 x^{14} d x = \frac{108 x ^{15}}{5}

\int 27 x^{9} d x = \frac{27 x ^{10}}{10}

\int x^{4} d x = \frac{x ^{5}}{5}

= \frac{19683 x ^{50}}{50} + \frac{6561 x ^{45}}{5} + \frac{19683 x ^{40}}{10} + \frac{8748 x ^{35}}{5} + \frac{5103 x ^{30}}{5} + \frac{10206 x ^{25}}{25} + \frac{567 x ^{20}}{5} + \frac{108 x ^{15}}{5} + \frac{27 x ^{10}}{10} + \frac{x ^{5}}{5}

定数を解答に追加する

= \frac{19683 x ^{50}}{50} + \frac{6561 x ^{45}}{5} + \frac{19683 x ^{40}}{10} + \frac{8748 x ^{35}}{5} + \frac{5103 x ^{30}}{5} + \frac{10206 x ^{25}}{25} + \frac{567 x ^{20}}{5} + \frac{108 x ^{15}}{5} + \frac{27 x ^{10}}{10} + \frac{x ^{5}}{5} + C