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受欢迎的微积分 >

integral from 0 to 2 of 8e^{2x}

解答

\int_{0}^{2} 8 e^{2 x} d x

解答

4 (e^{4} - 1)

+1

十进制

214.39260 \dots

求解步骤

\int_{0}^{2} 8 e^{2 x} d x

提出常数:

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

= 8 \cdot \int_{0}^{2} e^{2 x} d x

使用换元积分法

= 8 \cdot \int_{0}^{4} e^{u} \frac{1}{2} d u

提出常数:

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

= 8 \cdot \frac{1}{2} \cdot \int_{0}^{4} e^{u} d u

使用积分常见定则:

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

= 8 \cdot \frac{1}{2} [e^{u}]_{0}^{4}

化简

8 \cdot \frac{1}{2} [e^{u}]_{0}^{4} : 4 [e^{u}]_{0}^{4}

= 4 [e^{u}]_{0}^{4}

带入上下限计算后得:

e^{4} - 1

= 4 (e^{4} - 1)

作图

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