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人気のある微分積分 >

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

解

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

解

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

解答ステップ

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

u置換積分法を適用する

= \int - \frac{u ^{\frac{3}{2}}}{2} d u

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \frac{1}{2} \cdot \int u^{\frac{3}{2}} d u

べき乗則を適用する

= - \frac{1}{2} \cdot \frac{2}{5} u^{\frac{5}{2}}

代用を戻す

u = 1 - x^{2}

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

簡素化

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

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

定数を解答に追加する

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

グラフ

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