integral of x^3(16-x^4)^5

解

\int x^{3} (16 - x^{4})^{5} d x

262144 x^{4} - 40960 x^{8} + \frac{10240 x ^{12}}{3} - 160 x^{16} + 4 x^{20} - \frac{x ^{24}}{24} + C

解答ステップ

\int x^{3} (16 - x^{4})^{5} d x

拡張

x^{3} (16 - x^{4})^{5} : 1048576 x^{3} - 327680 x^{7} + 40960 x^{11} - 2560 x^{15} + 80 x^{19} - x^{23}

= \int 1048576 x^{3} - 327680 x^{7} + 40960 x^{11} - 2560 x^{15} + 80 x^{19} - x^{23} 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 1048576 x^{3} d x - \int 327680 x^{7} d x + \int 40960 x^{11} d x - \int 2560 x^{15} d x + \int 80 x^{19} d x - \int x^{23} d x

\int 1048576 x^{3} d x = 262144 x^{4}

\int 327680 x^{7} d x = 40960 x^{8}

\int 40960 x^{11} d x = \frac{10240 x ^{12}}{3}

\int 2560 x^{15} d x = 160 x^{16}

\int 80 x^{19} d x = 4 x^{20}

\int x^{23} d x = \frac{x ^{24}}{24}

= 262144 x^{4} - 40960 x^{8} + \frac{10240 x ^{12}}{3} - 160 x^{16} + 4 x^{20} - \frac{x ^{24}}{24}

定数を解答に追加する

= 262144 x^{4} - 40960 x^{8} + \frac{10240 x ^{12}}{3} - 160 x^{16} + 4 x^{20} - \frac{x ^{24}}{24} + C