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人気のある微分積分 >

integral of sin^4(2x)cos^6(2x)

解

\int sin^{4} (2 x) cos^{6} (2 x) d x

解

\frac{1}{5120} (60 x + 16 cos^{5} (2 x) sin (2 x) + 20 cos^{3} (2 x) sin (2 x) + 15 sin (4 x) - 352 cos^{7} (2 x) sin (2 x) + 256 cos^{9} (2 x) sin (2 x)) + C

解答ステップ

\int sin^{4} (2 x) cos^{6} (2 x) d x

u置換積分法を適用する

= \int sin^{4} (u) cos^{6} (u) \frac{1}{2} d u

定数を除く:

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

= \frac{1}{2} \cdot \int sin^{4} (u) cos^{6} (u) d u

三角関数の公式を使用して書き換える

= \frac{1}{2} \cdot \int (1 - cos^{2} (u))^{2} cos^{6} (u) d u

拡張

(1 - cos^{2} (u))^{2} cos^{6} (u) : cos^{6} (u) - 2 cos^{8} (u) + cos^{10} (u)

= \frac{1}{2} \cdot \int cos^{6} (u) - 2 cos^{8} (u) + cos^{10} (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

= \frac{1}{2} (\int cos^{6} (u) d u - \int 2 cos^{8} (u) d u + \int cos^{10} (u) d u)

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

\int 2 cos^{8} (u) d u = 2 (\frac{sin ( u ) cos ^{7} ( u )}{8} + \frac{7}{8} (\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)))))

\int cos^{10} (u) d u = \frac{sin ( u ) cos ^{9} ( u )}{10} + \frac{9}{10} (\frac{sin ( u ) cos ^{7} ( u )}{8} + \frac{7}{8} (\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)))))

= \frac{1}{2} (\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))) - 2 (\frac{sin ( u ) cos ^{7} ( u )}{8} + \frac{7}{8} (\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))))) + \frac{sin ( u ) cos ^{9} ( u )}{10} + \frac{9}{10} (\frac{sin ( u ) cos ^{7} ( u )}{8} + \frac{7}{8} (\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 = 2 x

= \frac{1}{2} (\frac{sin ( 2 x ) cos ^{5} ( 2 x )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (2 x) sin (2 x) + \frac{3}{8} (2 x + \frac{1}{2} sin (2 \cdot 2 x))) - 2 (\frac{sin ( 2 x ) cos ^{7} ( 2 x )}{8} + \frac{7}{8} (\frac{sin ( 2 x ) cos ^{5} ( 2 x )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (2 x) sin (2 x) + \frac{3}{8} (2 x + \frac{1}{2} sin (2 \cdot 2 x))))) + \frac{sin ( 2 x ) cos ^{9} ( 2 x )}{10} + \frac{9}{10} (\frac{sin ( 2 x ) cos ^{7} ( 2 x )}{8} + \frac{7}{8} (\frac{sin ( 2 x ) cos ^{5} ( 2 x )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (2 x) sin (2 x) + \frac{3}{8} (2 x + \frac{1}{2} sin (2 \cdot 2 x))))))

簡素化

\frac{1}{2} (\frac{sin ( 2 x ) cos ^{5} ( 2 x )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (2 x) sin (2 x) + \frac{3}{8} (2 x + \frac{1}{2} sin (2 \cdot 2 x))) - 2 (\frac{sin ( 2 x ) cos ^{7} ( 2 x )}{8} + \frac{7}{8} (\frac{sin ( 2 x ) cos ^{5} ( 2 x )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (2 x) sin (2 x) + \frac{3}{8} (2 x + \frac{1}{2} sin (2 \cdot 2 x))))) + \frac{sin ( 2 x ) cos ^{9} ( 2 x )}{10} + \frac{9}{10} (\frac{sin ( 2 x ) cos ^{7} ( 2 x )}{8} + \frac{7}{8} (\frac{sin ( 2 x ) cos ^{5} ( 2 x )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (2 x) sin (2 x) + \frac{3}{8} (2 x + \frac{1}{2} sin (2 \cdot 2 x)))))) : \frac{1}{5120} (60 x + 16 cos^{5} (2 x) sin (2 x) + 20 cos^{3} (2 x) sin (2 x) + 15 sin (4 x) - 352 cos^{7} (2 x) sin (2 x) + 256 cos^{9} (2 x) sin (2 x))

= \frac{1}{5120} (60 x + 16 cos^{5} (2 x) sin (2 x) + 20 cos^{3} (2 x) sin (2 x) + 15 sin (4 x) - 352 cos^{7} (2 x) sin (2 x) + 256 cos^{9} (2 x) sin (2 x))

定数を解答に追加する

= \frac{1}{5120} (60 x + 16 cos^{5} (2 x) sin (2 x) + 20 cos^{3} (2 x) sin (2 x) + 15 sin (4 x) - 352 cos^{7} (2 x) sin (2 x) + 256 cos^{9} (2 x) sin (2 x)) + C

グラフ

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