integral of x^3(x^2-4)^7

解

\int x^{3} (x^{2} - 4)^{7} d x

\frac{x ^{18}}{18} - \frac{7 x ^{16}}{4} + 24 x^{14} - \frac{560 x ^{12}}{3} + 896 x^{10} - 2688 x^{8} + \frac{14336 x ^{6}}{3} - 4096 x^{4} + C

解答ステップ

\int x^{3} (x^{2} - 4)^{7} d x

拡張

x^{3} (x^{2} - 4)^{7} : x^{17} - 28 x^{15} + 336 x^{13} - 2240 x^{11} + 8960 x^{9} - 21504 x^{7} + 28672 x^{5} - 16384 x^{3}

= \int x^{17} - 28 x^{15} + 336 x^{13} - 2240 x^{11} + 8960 x^{9} - 21504 x^{7} + 28672 x^{5} - 16384 x^{3} 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 x^{17} d x - \int 28 x^{15} d x + \int 336 x^{13} d x - \int 2240 x^{11} d x + \int 8960 x^{9} d x - \int 21504 x^{7} d x + \int 28672 x^{5} d x - \int 16384 x^{3} d x

\int x^{17} d x = \frac{x ^{18}}{18}

\int 28 x^{15} d x = \frac{7 x ^{16}}{4}

\int 336 x^{13} d x = 24 x^{14}

\int 2240 x^{11} d x = \frac{560 x ^{12}}{3}

\int 8960 x^{9} d x = 896 x^{10}

\int 21504 x^{7} d x = 2688 x^{8}

\int 28672 x^{5} d x = \frac{14336 x ^{6}}{3}

\int 16384 x^{3} d x = 4096 x^{4}

= \frac{x ^{18}}{18} - \frac{7 x ^{16}}{4} + 24 x^{14} - \frac{560 x ^{12}}{3} + 896 x^{10} - 2688 x^{8} + \frac{14336 x ^{6}}{3} - 4096 x^{4}

定数を解答に追加する

= \frac{x ^{18}}{18} - \frac{7 x ^{16}}{4} + 24 x^{14} - \frac{560 x ^{12}}{3} + 896 x^{10} - 2688 x^{8} + \frac{14336 x ^{6}}{3} - 4096 x^{4} + C