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受欢迎的微积分 >

integral of 3xe^{(x^2)/2}

解答

\int 3 x e^{\frac{x ^{2}}{2}} d x

解答

3 e^{\frac{x ^{2}}{2}} + C

求解步骤

\int 3 x e^{\frac{x ^{2}}{2}} d x

提出常数:

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

= 3 \cdot \int x e^{\frac{x ^{2}}{2}} d x

使用换元积分法

= 3 \cdot \int \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

= 3 \cdot \frac{1}{2} \cdot \int e^{\frac{u}{2}} d u

使用换元积分法

= 3 \cdot \frac{1}{2} \cdot \int e^{v} \cdot 2 d v

提出常数:

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

= 3 \cdot \frac{1}{2} \cdot 2 \cdot \int e^{v} d v

使用积分常见定则:

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

= 3 \cdot \frac{1}{2} \cdot 2 e^{v}

代回

= 3 \cdot \frac{1}{2} \cdot 2 e^{\frac{x ^{2}}{2}}

化简

3 \cdot \frac{1}{2} \cdot 2 e^{\frac{x ^{2}}{2}} : 3 e^{\frac{x ^{2}}{2}}

= 3 e^{\frac{x ^{2}}{2}}

解答补常数

= 3 e^{\frac{x ^{2}}{2}} + C

作图

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