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人気のある微分積分 >

integral from 0 to pi of cos^6(x/2)

解

\int_{0}^{π} cos^{6} (\frac{x}{2}) d x

解

\frac{5 π}{16}

+1

十進法表記

0.98174 \dots

解答ステップ

\int_{0}^{π} cos^{6} (\frac{x}{2}) d x

u置換積分法を適用する

= \int_{0}^{\frac{π}{2}} cos^{6} (u) \cdot 2 d u

定数を除く:

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

= 2 \cdot \int_{0}^{\frac{π}{2}} cos^{6} (u) d u

積分換算を適用する

= 2 ([\frac{sin ( u ) cos ^{5} ( u )}{6}]_{0}^{\frac{π}{2}} + \frac{5}{6} \cdot \int_{0}^{\frac{π}{2}} cos^{4} (u) d u)

\int_{0}^{\frac{π}{2}} cos^{4} (u) d u = \frac{3 π}{16}

= 2 ([\frac{sin ( u ) cos ^{5} ( u )}{6}]_{0}^{\frac{π}{2}} + \frac{5}{6} \cdot \frac{3 π}{16})

簡素化

2 ([\frac{sin ( u ) cos ^{5} ( u )}{6}]_{0}^{\frac{π}{2}} + \frac{5}{6} \cdot \frac{3 π}{16}) : 2 ([\frac{1}{6} cos^{5} (u) sin (u)]_{0}^{\frac{π}{2}} + \frac{5 π}{32})

= 2 ([\frac{1}{6} cos^{5} (u) sin (u)]_{0}^{\frac{π}{2}} + \frac{5 π}{32})

限度を計算する:

0

= 2 (0 + \frac{5 π}{32})

簡素化

= \frac{5 π}{16}

グラフ

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