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人気のある微分積分 >

integral from 0 to pi of 2xsin(3x)

解

\int_{0}^{π} 2 x sin (3 x) d x

解

\frac{2 π}{3}

+1

十進法表記

2.09439 \dots

解答ステップ

\int_{0}^{π} 2 x sin (3 x) d x

定数を除く:

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

= 2 \cdot \int_{0}^{π} x sin (3 x) d x

部分積分を適用する

= 2 [\frac{1}{3} (- x cos (3 x) - 3 \cdot \int - \frac{1}{3} cos (3 x) d x)]_{0}^{π}

\int - \frac{1}{3} cos (3 x) d x = - \frac{1}{9} sin (3 x)

= 2 [\frac{1}{3} (- x cos (3 x) - 3 (- \frac{1}{9} sin (3 x)))]_{0}^{π}

簡素化

2 [\frac{1}{3} (- x cos (3 x) - 3 (- \frac{1}{9} sin (3 x)))]_{0}^{π} : 2 [\frac{1}{3} (- x cos (3 x) + \frac{1}{3} sin (3 x))]_{0}^{π}

= 2 [\frac{1}{3} (- x cos (3 x) + \frac{1}{3} sin (3 x))]_{0}^{π}

限度を計算する:

\frac{π}{3}

= 2 \cdot \frac{π}{3}

簡素化

= \frac{2 π}{3}

グラフ

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