integral of (1-x^2)^2x^2

解

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

\frac{x ^{3}}{3} - \frac{2 x ^{5}}{5} + \frac{x ^{7}}{7} + C

解答ステップ

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

拡張

(1 - x^{2})^{2} x^{2} : x^{2} - 2 x^{4} + x^{6}

= \int x^{2} - 2 x^{4} + x^{6} 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^{2} d x - \int 2 x^{4} d x + \int x^{6} d x

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

\int 2 x^{4} d x = \frac{2 x ^{5}}{5}

\int x^{6} d x = \frac{x ^{7}}{7}

= \frac{x ^{3}}{3} - \frac{2 x ^{5}}{5} + \frac{x ^{7}}{7}

定数を解答に追加する

= \frac{x ^{3}}{3} - \frac{2 x ^{5}}{5} + \frac{x ^{7}}{7} + C