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受欢迎的微积分 >

integral from 0 to 1 of (1-x^2)^{(-1)/2}

解答

\int_{0}^{1} (1 - x^{2})^{\frac{- 1}{2}} d x

解答

\frac{π}{2}

+1

十进制

1.57079 \dots

求解步骤

\int_{0}^{1} (1 - x^{2})^{\frac{- 1}{2}} d x

化简

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

= \int_{0}^{1} \frac{1}{( 1 - x ^{2} ) ^{\frac{1}{2}}} d x

使用指数法则:

\frac{1}{a ^{b}} = a^{- b}

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

= \int_{0}^{1} (1 - x^{2})^{- \frac{1}{2}} d x

使用三角换元法

= \int_{0}^{\frac{π}{2}} 1 d u

常数积分:

\int a d x = a x

= [1 \cdot u]_{0}^{\frac{π}{2}}

化简

= [u]_{0}^{\frac{π}{2}}

带入上下限计算后得:

\frac{π}{2}

= \frac{π}{2}

作图

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