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受欢迎的微积分 >

integral from 1 to 4 of ((1+sqrt(x))^{1/2})/(sqrt(x))

解答

\int_{1}^{4} \frac{( 1 + x ) ^{\frac{1}{2}}}{x} d x

解答

2 (23 - \frac{4 2}{3})

+1

十进制

3.15696 \dots

求解步骤

\int_{1}^{4} \frac{( 1 + x ) ^{\frac{1}{2}}}{x} d x

化简

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

= \int_{1}^{4} \frac{1 + x}{x} d x

化简

\frac{1 + x}{x} : \frac{1 + x}{x}

= \int_{1}^{4} \frac{1 + x}{x} d x

使用换元积分法

= \int_{2}^{3} 2 u d u

提出常数:

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

= 2 \cdot \int_{2}^{3} u d u

使用幂法则

= 2 [\frac{2}{3} u^{\frac{3}{2}}]_{2}^{3}

带入上下限计算后得:

23 - \frac{4 2}{3}

= 2 (23 - \frac{4 2}{3})

作图

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