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受欢迎的微积分 >

integral from-1 to 1 of (x^2+6x)^3(2x+6)

解答

\int_{- 1}^{1} (x^{2} + 6 x)^{3} (2 x + 6) d x

解答

444

求解步骤

\int_{- 1}^{1} (x^{2} + 6 x)^{3} (2 x + 6) d x

使用换元积分法

= \int_{4}^{8} \frac{u ( u ^{2} - 36 ) ^{3}}{128} d u

提出常数:

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

= \frac{1}{128} \cdot \int_{4}^{8} u (u^{2} - 36)^{3} d u

使用换元积分法

= \frac{1}{128} \cdot \int_{- 20}^{28} \frac{v ^{3}}{2} d v

提出常数:

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

= \frac{1}{128} \cdot \frac{1}{2} \cdot \int_{- 20}^{28} v^{3} d v

使用幂法则

= \frac{1}{128} \cdot \frac{1}{2} [\frac{v ^{4}}{4}]_{- 20}^{28}

化简

\frac{1}{128} \cdot \frac{1}{2} [\frac{v ^{4}}{4}]_{- 20}^{28} : \frac{1}{256} [\frac{v ^{4}}{4}]_{- 20}^{28}

= \frac{1}{256} [\frac{v ^{4}}{4}]_{- 20}^{28}

带入上下限计算后得:

113664

= \frac{1}{256} \cdot 113664

化简

= 444

作图

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