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

解答

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

4 ln (2) - \frac{15}{16}

十进制

1.83508 \dots

求解步骤

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

使用分布积分法

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

\int \frac{x ^{3}}{4} d x = \frac{x ^{4}}{16}

= [\frac{1}{4} x^{4} ln (x) - \frac{x ^{4}}{16}]_{1}^{2}

带入上下限计算后得:

4 ln (2) - \frac{15}{16}

= 4 ln (2) - \frac{15}{16}

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