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人気のある微分積分 >

integral from 0 to pi/6 of cos^8(3x)

解

\int_{0}^{\frac{π}{6}} cos^{8} (3 x) d x

解

\frac{35 π}{768}

+1

十進法表記

0.14317 \dots

解答ステップ

\int_{0}^{\frac{π}{6}} cos^{8} (3 x) d x

u置換積分法を適用する

= \int_{0}^{\frac{π}{2}} cos^{8} (u) \frac{1}{3} d u

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}^{\frac{π}{2}} cos^{8} (u) d u

積分換算を適用する

= \frac{1}{3} ([\frac{sin ( u ) cos ^{7} ( u )}{8}]_{0}^{\frac{π}{2}} + \frac{7}{8} \cdot \int_{0}^{\frac{π}{2}} cos^{6} (u) d u)

\int_{0}^{\frac{π}{2}} cos^{6} (u) d u = \frac{5 π}{32}

= \frac{1}{3} ([\frac{sin ( u ) cos ^{7} ( u )}{8}]_{0}^{\frac{π}{2}} + \frac{7}{8} \cdot \frac{5 π}{32})

簡素化

\frac{1}{3} ([\frac{sin ( u ) cos ^{7} ( u )}{8}]_{0}^{\frac{π}{2}} + \frac{7}{8} \cdot \frac{5 π}{32}) : \frac{1}{3} ([\frac{1}{8} cos^{7} (u) sin (u)]_{0}^{\frac{π}{2}} + \frac{35 π}{256})

= \frac{1}{3} ([\frac{1}{8} cos^{7} (u) sin (u)]_{0}^{\frac{π}{2}} + \frac{35 π}{256})

限度を計算する:

0

= \frac{1}{3} (0 + \frac{35 π}{256})

簡素化

= \frac{35 π}{768}

グラフ

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