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受欢迎的微积分 >

integral from 0 to 2 of 15e^{3x}

解答

\int_{0}^{2} 15 e^{3 x} d x

解答

5 (e^{6} - 1)

+1

十进制

2012.14396 \dots

求解步骤

\int_{0}^{2} 15 e^{3 x} d x

提出常数:

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

= 15 \cdot \int_{0}^{2} e^{3 x} d x

使用换元积分法

= 15 \cdot \int_{0}^{6} e^{u} \frac{1}{3} d u

提出常数:

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

= 15 \cdot \frac{1}{3} \cdot \int_{0}^{6} e^{u} d u

使用积分常见定则:

\int e^{u} d u = e^{u}

= 15 \cdot \frac{1}{3} [e^{u}]_{0}^{6}

化简

15 \cdot \frac{1}{3} [e^{u}]_{0}^{6} : 5 [e^{u}]_{0}^{6}

= 5 [e^{u}]_{0}^{6}

带入上下限计算后得:

e^{6} - 1

= 5 (e^{6} - 1)

作图

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