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人気のある微分積分 >

integral from 0 to 2pi of 1/3 cos^4(x)

解

\int_{0}^{2 π} \frac{1}{3} cos^{4} (x) d x

解

\frac{π}{4}

+1

十進法表記

0.78539 \dots

解答ステップ

\int_{0}^{2 π} \frac{1}{3} cos^{4} (x) d x

定数を除く:

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

= \frac{1}{3} \cdot \int_{0}^{2 π} cos^{4} (x) d x

簡素化

cos^{4} (x) : cos^{3} (x) cos (x)

= \frac{1}{3} \cdot \int_{0}^{2 π} cos^{3} (x) cos (x) d x

部分積分を適用する

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

\int - 3 cos^{2} (x) sin^{2} (x) d x = - \frac{3}{8} (x - \frac{1}{4} sin (4 x))

= \frac{1}{3} [cos^{3} (x) sin (x) - (- \frac{3}{8} (x - \frac{1}{4} sin (4 x)))]_{0}^{2 π}

簡素化

\frac{1}{3} [cos^{3} (x) sin (x) - (- \frac{3}{8} (x - \frac{1}{4} sin (4 x)))]_{0}^{2 π} : \frac{1}{3} [\frac{1}{32} (32 cos^{3} (x) sin (x) + 3 (4 x - sin (4 x)))]_{0}^{2 π}

= \frac{1}{3} [\frac{1}{32} (32 cos^{3} (x) sin (x) + 3 (4 x - sin (4 x)))]_{0}^{2 π}

限度を計算する:

\frac{3 π}{4}

= \frac{1}{3} \cdot \frac{3 π}{4}

簡素化

= \frac{π}{4}

グラフ

人気の例

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