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受欢迎的微积分 >

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

解答

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

解答

4

求解步骤

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

提出常数:

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

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

使用换元积分法

= 9 \cdot \int_{0}^{2} \frac{u ^{2}}{6} d u

提出常数:

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

= 9 \cdot \frac{1}{6} \cdot \int_{0}^{2} u^{2} d u

使用幂法则

= 9 \cdot \frac{1}{6} [\frac{u ^{3}}{3}]_{0}^{2}

化简

9 \cdot \frac{1}{6} [\frac{u ^{3}}{3}]_{0}^{2} : \frac{3}{2} [\frac{u ^{3}}{3}]_{0}^{2}

= \frac{3}{2} [\frac{u ^{3}}{3}]_{0}^{2}

带入上下限计算后得:

\frac{8}{3}

= \frac{3}{2} \cdot \frac{8}{3}

化简

= 4

作图

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