integral of x((1-x^2)^{3/2}-x^3)

解

\int x ((1 - x^{2})^{\frac{3}{2}} - x^{3}) d x

- \frac{1}{5} (1 - x^{2})^{\frac{5}{2}} - \frac{x ^{5}}{5} + C

解答ステップ

\int x ((1 - x^{2})^{\frac{3}{2}} - x^{3}) d x

拡張

x ((1 - x^{2})^{\frac{3}{2}} - x^{3}) : x (1 - x^{2})^{\frac{3}{2}} - x^{4}

= \int x (1 - x^{2})^{\frac{3}{2}} - 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 x (1 - x^{2})^{\frac{3}{2}} d x - \int x^{4} d x

\int x (1 - x^{2})^{\frac{3}{2}} d x = - \frac{1}{5} (1 - x^{2})^{\frac{5}{2}}

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

= - \frac{1}{5} (1 - x^{2})^{\frac{5}{2}} - \frac{x ^{5}}{5}

定数を解答に追加する

= - \frac{1}{5} (1 - x^{2})^{\frac{5}{2}} - \frac{x ^{5}}{5} + C