integral from 1 to 2 of 7r^2ln(r)

解答

\int_{1}^{2} 7 r^{2} ln (r) d r

7 (- \frac{7}{9} + \frac{8}{3} ln (2))

十进制

7.49430 \dots

求解步骤

\int_{1}^{2} 7 r^{2} ln (r) d r

提出常数:

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

= 7 \cdot \int_{1}^{2} r^{2} ln (r) d r

使用分布积分法

= 7 [\frac{1}{3} r^{3} ln (r) - \int \frac{r ^{2}}{3} d r]_{1}^{2}

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

= 7 [\frac{1}{3} r^{3} ln (r) - \frac{r ^{3}}{9}]_{1}^{2}

带入上下限计算后得:

- \frac{7}{9} + \frac{8}{3} ln (2)

= 7 (- \frac{7}{9} + \frac{8}{3} ln (2))

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