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

解答

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

\frac{3 π}{8}

十进制

1.17809 \dots

求解步骤

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

使用分布积分法

= [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}

化简

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

带入上下限计算后得:

\frac{3 π}{8}

= \frac{3 π}{8}

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