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受欢迎的微积分 >

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

解答

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

解答

e^{2} - 1

+1

十进制

6.38905 \dots

求解步骤

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

使用换元积分法

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

使用换元积分法

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

提出常数:

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

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

使用积分常见定则:

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

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

化简

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

= [e^{v}]_{0}^{2}

带入上下限计算后得:

e^{2} - 1

= e^{2} - 1

作图

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