integral from 0 to 4 of 3x^2ln(2x)

解答

\int_{0}^{4} 3 x^{2} ln (2 x) d x

3 (64 ln (2) - \frac{64}{9})

十进制

111.75092 \dots

求解步骤

\int_{0}^{4} 3 x^{2} ln (2 x) d x

提出常数:

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

= 3 \cdot \int_{0}^{4} x^{2} ln (2 x) d x

使用分布积分法

= 3 [\frac{1}{3} x^{3} ln (2 x) - \int \frac{x ^{2}}{3} d x]_{0}^{4}

\int \frac{x ^{2}}{3} d x = \frac{x ^{3}}{9}

= 3 [\frac{1}{3} x^{3} ln (2 x) - \frac{x ^{3}}{9}]_{0}^{4}

带入上下限计算后得:

64 ln (2) - \frac{64}{9}

= 3 (64 ln (2) - \frac{64}{9})

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