integral from 1 to 5 of 7r^2ln(r)

解

\int_{1}^{5} 7 r^{2} ln (r) d r

7 (- \frac{124}{9} + \frac{125}{3} ln (5))

十進法表記

372.97494 \dots

解答ステップ

\int_{1}^{5} 7 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

= 7 \cdot \int_{1}^{5} r^{2} ln (r) d r

部分積分を適用する

= 7 [\frac{1}{3} r^{3} ln (r) - \int \frac{r ^{2}}{3} d r]_{1}^{5}

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

= 7 [\frac{1}{3} r^{3} ln (r) - \frac{r ^{3}}{9}]_{1}^{5}

限度を計算する:

- \frac{124}{9} + \frac{125}{3} ln (5)

= 7 (- \frac{124}{9} + \frac{125}{3} ln (5))