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受欢迎的微积分 >

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

解答

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

解答

\frac{52}{9}

+1

十进制

5.77777 \dots

求解步骤

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

使用换元积分法

= \int_{0}^{8} \frac{( u + 1 ) ^{\frac{1}{2}}}{3} d u

提出常数:

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

= \frac{1}{3} \cdot \int_{0}^{8} (u + 1)^{\frac{1}{2}} d u

使用换元积分法

= \frac{1}{3} \cdot \int_{1}^{9} v^{\frac{1}{2}} d v

使用幂法则

= \frac{1}{3} [\frac{2}{3} v^{\frac{3}{2}}]_{1}^{9}

带入上下限计算后得:

\frac{52}{3}

= \frac{1}{3} \cdot \frac{52}{3}

化简

= \frac{52}{9}

作图

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