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

解

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

\frac{3 π}{8}

十進法表記

1.17809 \dots

解答ステップ

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

部分積分を適用する

= [x (1 - x^{2})^{\frac{3}{2}} - \int - 3 x^{2} (1 - x^{2})^{\frac{1}{2}} d x]_{- 1}^{1}

\int - 3 x^{2} (1 - x^{2})^{\frac{1}{2}} d x = - \frac{3}{8} (arcsin (x) - \frac{1}{4} sin (4 arcsin (x)))

= [x (1 - x^{2})^{\frac{3}{2}} - (- \frac{3}{8} (arcsin (x) - \frac{1}{4} sin (4 arcsin (x))))]_{- 1}^{1}

簡素化

= [x (1 - x^{2})^{\frac{3}{2}} + \frac{3}{8} (arcsin (x) - \frac{1}{4} sin (4 arcsin (x)))]_{- 1}^{1}

限度を計算する:

\frac{3 π}{8}

= \frac{3 π}{8}