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人気のある微分積分 >

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

解

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

解

\frac{27 π}{2}

+1

十進法表記

42.41150 \dots

解答ステップ

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

定数を除く:

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

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

拡張

(3 + 3 cos (x))^{2} : 9 + 18 cos (x) + 9 cos^{2} (x)

= \frac{1}{2} \cdot \int_{0}^{2 π} 9 + 18 cos (x) + 9 cos^{2} (x) d x

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (\int_{0}^{2 π} 9 d x + \int_{0}^{2 π} 18 cos (x) d x + \int_{0}^{2 π} 9 cos^{2} (x) d x)

\int_{0}^{2 π} 9 d x = 18 π

\int_{0}^{2 π} 18 cos (x) d x = 0

\int_{0}^{2 π} 9 cos^{2} (x) d x = 9 π

= \frac{1}{2} (18 π + 0 + 9 π)

簡素化

\frac{1}{2} (18 π + 0 + 9 π) : \frac{27 π}{2}

= \frac{27 π}{2}

グラフ

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