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受欢迎的微积分 >

integral from 0 to 1 of (e^{2x})/2

解答

\int_{0}^{1} \frac{e ^{2 x}}{2} d x

解答

\frac{e ^{2} - 1}{4}

+1

十进制

1.59726 \dots

求解步骤

\int_{0}^{1} \frac{e ^{2 x}}{2} d x

提出常数:

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

= \frac{1}{2} \cdot \int_{0}^{1} e^{2 x} d x

使用换元积分法

= \frac{1}{2} \cdot \int_{0}^{2} 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

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

使用积分常见定则:

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

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

化简

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

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

带入上下限计算后得:

e^{2} - 1

= \frac{1}{4} (e^{2} - 1)

化简

= \frac{e ^{2} - 1}{4}

作图

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