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人気のある微分積分 >

integral from 0 to 2pi of tcos^2(t)

解

\int_{0}^{2 π} t cos^{2} (t) d t

解

π^{2}

+1

十進法表記

9.86960 \dots

解答ステップ

\int_{0}^{2 π} t cos^{2} (t) d t

部分積分を適用する

= [\frac{1}{4} (t (2 t + sin (2 t)) - 4 \cdot \int \frac{1}{2} (t + \frac{1}{2} sin (2 t)) d t)]_{0}^{2 π}

\int \frac{1}{2} (t + \frac{1}{2} sin (2 t)) d t = \frac{1}{2} (\frac{t ^{2}}{2} - \frac{1}{4} cos (2 t))

= [\frac{1}{4} (t (2 t + sin (2 t)) - 4 \cdot \frac{1}{2} (\frac{t ^{2}}{2} - \frac{1}{4} cos (2 t)))]_{0}^{2 π}

簡素化

[\frac{1}{4} (t (2 t + sin (2 t)) - 4 \cdot \frac{1}{2} (\frac{t ^{2}}{2} - \frac{1}{4} cos (2 t)))]_{0}^{2 π} : [\frac{1}{8} (2 t^{2} + 2 t sin (2 t) + cos (2 t))]_{0}^{2 π}

= [\frac{1}{8} (2 t^{2} + 2 t sin (2 t) + cos (2 t))]_{0}^{2 π}

限度を計算する:

\frac{8 π ^{2} + 1}{8} - \frac{1}{8}

= \frac{8 π ^{2} + 1}{8} - \frac{1}{8}

簡素化

= π^{2}

グラフ

人気の例

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