integral from 1/5 to 3 of 14xln(5x)

解

\int_{\frac{1}{5}}^{3} 14 x ln (5 x) d x

14 (- \frac{56}{25} + \frac{9}{2} ln (15))

十進法表記

139.24716 \dots

解答ステップ

\int_{\frac{1}{5}}^{3} 14 x ln (5 x) d x

定数を除く:

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

= 14 \cdot \int_{\frac{1}{5}}^{3} x ln (5 x) d x

部分積分を適用する

= 14 [\frac{1}{2} x^{2} ln (5 x) - \int \frac{x}{2} d x]_{\frac{1}{5}}^{3}

\int \frac{x}{2} d x = \frac{x ^{2}}{4}

= 14 [\frac{1}{2} x^{2} ln (5 x) - \frac{x ^{2}}{4}]_{\frac{1}{5}}^{3}

限度を計算する:

- \frac{56}{25} + \frac{9}{2} ln (15)

= 14 (- \frac{56}{25} + \frac{9}{2} ln (15))