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人気のある微分積分 >

integral from 0 to pi of pi(5sin^2(x))^2

解

\int_{0}^{π} π (5 sin^{2} (x))^{2} d x

解

\frac{75 π ^{2}}{8}

+1

十進法表記

92.52754 \dots

解答ステップ

\int_{0}^{π} π (5 sin^{2} (x))^{2} d x

定数を除く:

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

= π \cdot \int_{0}^{π} (5 sin^{2} (x))^{2} d x

三角関数の公式を使用して書き換える

= π \cdot \int_{0}^{π} \frac{25}{4} (1 - cos (2 x))^{2} d x

定数を除く:

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

= π \frac{25}{4} \cdot \int_{0}^{π} (1 - cos (2 x))^{2} d x

u置換積分法を適用する

= π \frac{25}{4} \cdot \int_{0}^{2 π} \frac{1}{2} - cos (u) + \frac{1}{2} cos^{2} (u) d u

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

= π \frac{25}{4} (\int_{0}^{2 π} \frac{1}{2} d u - \int_{0}^{2 π} cos (u) d u + \int_{0}^{2 π} \frac{1}{2} cos^{2} (u) d u)

\int_{0}^{2 π} \frac{1}{2} d u = π

\int_{0}^{2 π} cos (u) d u = 0

\int_{0}^{2 π} \frac{1}{2} cos^{2} (u) d u = \frac{π}{2}

= π \frac{25}{4} (π - 0 + \frac{π}{2})

簡素化

π \frac{25}{4} (π - 0 + \frac{π}{2}) : \frac{75 π ^{2}}{8}

= \frac{75 π ^{2}}{8}

グラフ

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