integral of (1-x^2)^8

解

\int (1 - x^{2})^{8} d x

x - \frac{8 x ^{3}}{3} + \frac{28 x ^{5}}{5} - 8 x^{7} + \frac{70 x ^{9}}{9} - \frac{56 x ^{11}}{11} + \frac{28 x ^{13}}{13} - \frac{8 x ^{15}}{15} + \frac{x ^{17}}{17} + C

解答ステップ

\int (1 - x^{2})^{8} d x

拡張

(1 - x^{2})^{8} : 1 - 8 x^{2} + 28 x^{4} - 56 x^{6} + 70 x^{8} - 56 x^{10} + 28 x^{12} - 8 x^{14} + x^{16}

= \int 1 - 8 x^{2} + 28 x^{4} - 56 x^{6} + 70 x^{8} - 56 x^{10} + 28 x^{12} - 8 x^{14} + x^{16} 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 1 d x - \int 8 x^{2} d x + \int 28 x^{4} d x - \int 56 x^{6} d x + \int 70 x^{8} d x - \int 56 x^{10} d x + \int 28 x^{12} d x - \int 8 x^{14} d x + \int x^{16} d x

\int 1 d x = x

\int 8 x^{2} d x = \frac{8 x ^{3}}{3}

\int 28 x^{4} d x = \frac{28 x ^{5}}{5}

\int 56 x^{6} d x = 8 x^{7}

\int 70 x^{8} d x = \frac{70 x ^{9}}{9}

\int 56 x^{10} d x = \frac{56 x ^{11}}{11}

\int 28 x^{12} d x = \frac{28 x ^{13}}{13}

\int 8 x^{14} d x = \frac{8 x ^{15}}{15}

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

= x - \frac{8 x ^{3}}{3} + \frac{28 x ^{5}}{5} - 8 x^{7} + \frac{70 x ^{9}}{9} - \frac{56 x ^{11}}{11} + \frac{28 x ^{13}}{13} - \frac{8 x ^{15}}{15} + \frac{x ^{17}}{17}

定数を解答に追加する

= x - \frac{8 x ^{3}}{3} + \frac{28 x ^{5}}{5} - 8 x^{7} + \frac{70 x ^{9}}{9} - \frac{56 x ^{11}}{11} + \frac{28 x ^{13}}{13} - \frac{8 x ^{15}}{15} + \frac{x ^{17}}{17} + C