integral of xe^{ax^2}

解答

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

\frac{1}{2 a} e^{a x^{2}} + C

求解步骤

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

使用换元积分法

= \int \frac{e ^{u}}{2 a} d u

提出常数:

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

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

使用积分常见定则:

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

= \frac{1}{2 a} e^{u}

u = a x^{2}

代回

= \frac{1}{2 a} e^{a x^{2}}

解答补常数

= \frac{1}{2 a} e^{a x^{2}} + C

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