integral from 1 to 2 of 8r^2ln(r)

解

\int_{1}^{2} 8 r^{2} ln (r) d r

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

十進法表記

8.56491 \dots

解答ステップ

\int_{1}^{2} 8 r^{2} ln (r) d r

定数を除く:

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

= 8 \cdot \int_{1}^{2} r^{2} ln (r) d r

部分積分を適用する

= 8 [\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}

= 8 [\frac{1}{3} r^{3} ln (r) - \frac{r ^{3}}{9}]_{1}^{2}

限度を計算する:

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

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