integral from 1 to 2 of x^2ln(x^3)

解答

\int_{1}^{2} x^{2} ln (x^{3}) d x

8 ln (2) - \frac{7}{3}

十进制

3.21184 \dots

求解步骤

\int_{1}^{2} x^{2} ln (x^{3}) d x

使用分布积分法

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

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

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

\frac{1}{3} x^{3} ln (x^{3}) = x^{3} ln (x)

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

带入上下限计算后得:

8 ln (2) - \frac{7}{3}

= 8 ln (2) - \frac{7}{3}

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