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受欢迎的微积分 >

integral from 0 to 2 of 9x^2sqrt(1+x^3)

解答

\int_{0}^{2} 9 x^{2} 1 + x^{3} d x

解答

52

求解步骤

\int_{0}^{2} 9 x^{2} 1 + x^{3} d x

提出常数:

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

= 9 \cdot \int_{0}^{2} x^{2} 1 + x^{3} d x

使用换元积分法

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

提出常数:

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

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

使用幂法则

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

化简

9 \cdot \frac{1}{3} [\frac{2}{3} u^{\frac{3}{2}}]_{1}^{9} : 3 [\frac{2}{3} u^{\frac{3}{2}}]_{1}^{9}

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

带入上下限计算后得:

\frac{52}{3}

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

化简

= 52

作图

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