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受欢迎的微积分 >

integral from 0 to ln(2) of 2e^{3x}

解答

\int_{0}^{l n (2)} 2 e^{3 x} d x

解答

\frac{14}{3}

+1

十进制

4.66666 \dots

求解步骤

\int_{0}^{l n (2)} 2 e^{3 x} d x

提出常数:

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

= 2 \cdot \int_{0}^{l n (2)} e^{3 x} d x

使用换元积分法

= 2 \cdot \int_{0}^{3 l n (2)} 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

= 2 \cdot \frac{1}{3} \cdot \int_{0}^{3 l n (2)} e^{u} d u

使用积分常见定则:

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

= 2 \cdot \frac{1}{3} [e^{u}]_{0}^{3 l n (2)}

化简

2 \cdot \frac{1}{3} [e^{u}]_{0}^{3 l n (2)} : \frac{2}{3} [e^{u}]_{0}^{3 l n (2)}

= \frac{2}{3} [e^{u}]_{0}^{3 l n (2)}

带入上下限计算后得:

7

= \frac{2}{3} \cdot 7

化简

= \frac{14}{3}

作图

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