integral from 0 to 3 of (3-x)^{(-1)/2}

解答

\int_{0}^{3} (3 - x)^{\frac{- 1}{2}} d x

2 \cdot 3^{\frac{1}{2}}

十进制

3.46410 \dots

求解步骤

\int_{0}^{3} (3 - x)^{\frac{- 1}{2}} d x

化简

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

= \int_{0}^{3} \frac{1}{( 3 - x ) ^{\frac{1}{2}}} d x

使用分布积分法

= [\frac{x}{( 3 - x ) ^{\frac{1}{2}}} - \int \frac{x}{2 ( 3 - x ) ^{\frac{3}{2}}} d x]_{0}^{3}

\int \frac{x}{2 ( 3 - x ) ^{\frac{3}{2}}} d x = - \frac{x - 6}{( 3 - x ) ^{\frac{1}{2}}}

= [\frac{x}{( 3 - x ) ^{\frac{1}{2}}} - (- \frac{x - 6}{( 3 - x ) ^{\frac{1}{2}}})]_{0}^{3}

化简

[\frac{x}{( 3 - x ) ^{\frac{1}{2}}} - (- \frac{x - 6}{( 3 - x ) ^{\frac{1}{2}}})]_{0}^{3} : [- 2 (- x + 3)^{\frac{1}{2}}]_{0}^{3}

= [- 2 (- x + 3)^{\frac{1}{2}}]_{0}^{3}

带入上下限计算后得:

2 \cdot 3^{\frac{1}{2}}

= 2 \cdot 3^{\frac{1}{2}}

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