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受欢迎的微积分 >

integral from 0 to 4 of (e^{y^2}y)/2

解答

\int_{0}^{4} \frac{e ^{y^{2}} y}{2} d y

解答

\frac{e ^{16} - 1}{4}

+1

十进制

2221527.38012 \dots

求解步骤

\int_{0}^{4} \frac{e ^{y^{2}} y}{2} d y

提出常数:

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

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

使用换元积分法

= \frac{1}{2} \cdot \int_{0}^{16} \frac{e ^{u}}{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}^{16} 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}^{16}

化简

\frac{1}{2} \cdot \frac{1}{2} [e^{u}]_{0}^{16} : \frac{1}{4} [e^{u}]_{0}^{16}

= \frac{1}{4} [e^{u}]_{0}^{16}

带入上下限计算后得:

e^{16} - 1

= \frac{1}{4} (e^{16} - 1)

化简

= \frac{e ^{16} - 1}{4}

作图

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