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受欢迎的微积分 >

integral of 15xe^{x^2}

解答

\int 15 x e^{x^{2}} d x

解答

\frac{15}{2} e^{x^{2}} + C

求解步骤

\int 15 x e^{x^{2}} d x

提出常数:

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

= 15 \cdot \int x e^{x^{2}} d x

使用换元积分法

= 15 \cdot \int \frac{e ^{u}}{2} d u

提出常数:

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

= 15 \cdot \frac{1}{2} \cdot \int e^{u} d u

使用积分常见定则:

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

= 15 \cdot \frac{1}{2} e^{u}

u = x^{2}

代回

= 15 \cdot \frac{1}{2} e^{x^{2}}

化简

15 \cdot \frac{1}{2} e^{x^{2}} : \frac{15}{2} e^{x^{2}}

= \frac{15}{2} e^{x^{2}}

解答补常数

= \frac{15}{2} e^{x^{2}} + C

作图

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