integral from 0 to 1/2 of (xe^{3x})^2

解答

\int_{0}^{\frac{1}{2}} (x e^{3 x})^{2} d x

\frac{5 e ^{3}}{216} - \frac{1}{108}

十进制

0.45568 \dots

求解步骤

\int_{0}^{\frac{1}{2}} (x e^{3 x})^{2} d x

化简

(x e^{3 x})^{2} : e^{6 x} x^{2}

= \int_{0}^{\frac{1}{2}} e^{6 x} x^{2} d x

使用分布积分法

= [\frac{1}{6} (e^{6 x} x^{2} - 6 \cdot \int \frac{1}{3} e^{6 x} x d x)]_{0}^{\frac{1}{2}}

\int \frac{1}{3} e^{6 x} x d x = \frac{1}{108} (6 e^{6 x} x - e^{6 x})

= [\frac{1}{6} (e^{6 x} x^{2} - 6 \cdot \frac{1}{108} (6 e^{6 x} x - e^{6 x}))]_{0}^{\frac{1}{2}}

6 \cdot \frac{1}{108} (6 e^{6 x} x - e^{6 x}) = \frac{1}{18} (6 e^{6 x} x - e^{6 x})

= [\frac{1}{6} (e^{6 x} x^{2} - \frac{1}{18} (6 e^{6 x} x - e^{6 x}))]_{0}^{\frac{1}{2}}

带入上下限计算后得:

\frac{5 e ^{3}}{216} - \frac{1}{108}

= \frac{5 e ^{3}}{216} - \frac{1}{108}

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