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人気のある微分積分 >

integral of cos^6(3x)

解

\int cos^{6} (3 x) d x

解

\frac{1}{3} (\frac{1}{6} cos^{5} (3 x) sin (3 x) + \frac{5}{6} (\frac{1}{4} cos^{3} (3 x) sin (3 x) + \frac{3}{8} (3 x + \frac{1}{2} sin (6 x)))) + C

解答ステップ

\int cos^{6} (3 x) d x

u置換積分法を適用する

= \int cos^{6} (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 cos^{6} (u) d u

積分換算を適用する

= \frac{1}{3} (\frac{sin ( u ) cos ^{5} ( u )}{6} + \frac{5}{6} \cdot \int cos^{4} (u) d u)

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

= \frac{1}{3} (\frac{sin ( u ) cos ^{5} ( u )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (u) sin (u) + \frac{3}{8} (u + \frac{1}{2} sin (2 u))))

代用を戻す

u = 3 x

= \frac{1}{3} (\frac{sin ( 3 x ) cos ^{5} ( 3 x )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (3 x) sin (3 x) + \frac{3}{8} (3 x + \frac{1}{2} sin (2 \cdot 3 x))))

簡素化

\frac{1}{3} (\frac{sin ( 3 x ) cos ^{5} ( 3 x )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (3 x) sin (3 x) + \frac{3}{8} (3 x + \frac{1}{2} sin (2 \cdot 3 x)))) : \frac{1}{3} (\frac{1}{6} cos^{5} (3 x) sin (3 x) + \frac{5}{6} (\frac{1}{4} cos^{3} (3 x) sin (3 x) + \frac{3}{8} (3 x + \frac{1}{2} sin (6 x))))

= \frac{1}{3} (\frac{1}{6} cos^{5} (3 x) sin (3 x) + \frac{5}{6} (\frac{1}{4} cos^{3} (3 x) sin (3 x) + \frac{3}{8} (3 x + \frac{1}{2} sin (6 x))))

定数を解答に追加する

= \frac{1}{3} (\frac{1}{6} cos^{5} (3 x) sin (3 x) + \frac{5}{6} (\frac{1}{4} cos^{3} (3 x) sin (3 x) + \frac{3}{8} (3 x + \frac{1}{2} sin (6 x)))) + C

グラフ

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