integral from 0 to pi/4 of xsin(12x)

解

\int_{0}^{\frac{π}{4}} x sin (12 x) d x

\frac{π}{48}

十進法表記

0.06544 \dots

解答ステップ

\int_{0}^{\frac{π}{4}} x sin (12 x) d x

部分積分を適用する

= [\frac{1}{12} (- x cos (12 x) - 12 \cdot \int - \frac{1}{12} cos (12 x) d x)]_{0}^{\frac{π}{4}}

\int - \frac{1}{12} cos (12 x) d x = - \frac{1}{144} sin (12 x)

= [\frac{1}{12} (- x cos (12 x) - 12 (- \frac{1}{144} sin (12 x)))]_{0}^{\frac{π}{4}}

簡素化

[\frac{1}{12} (- x cos (12 x) - 12 (- \frac{1}{144} sin (12 x)))]_{0}^{\frac{π}{4}} : [\frac{1}{12} (- x cos (12 x) + \frac{1}{12} sin (12 x))]_{0}^{\frac{π}{4}}

= [\frac{1}{12} (- x cos (12 x) + \frac{1}{12} sin (12 x))]_{0}^{\frac{π}{4}}

限度を計算する:

\frac{π}{48}

= \frac{π}{48}