integral of 4x^2e^{3x^2}

解答

\int 4 x^{2} e^{3 x^{2}} d x

4 (\frac{1}{6} e^{3 x^{2}} x - \frac{π}{12 3} erfi (3 x)) + C

求解步骤

\int 4 x^{2} e^{3 x^{2}} d x

提出常数:

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

= 4 \cdot \int x^{2} e^{3 x^{2}} d x

化简

x^{2} : xx

= 4 \cdot \int xx e^{3 x^{2}} d x

使用分布积分法

= 4 (\frac{1}{6} e^{3 x^{2}} x - \int \frac{1}{6} e^{3 x^{2}} d x)

\int \frac{1}{6} e^{3 x^{2}} d x = \frac{π}{12 3} erfi (3 x)

= 4 (\frac{1}{6} e^{3 x^{2}} x - \frac{π}{12 3} erfi (3 x))

解答补常数

= 4 (\frac{1}{6} e^{3 x^{2}} x - \frac{π}{12 3} erfi (3 x)) + C

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