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

解

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

\frac{1}{2} x cos^{2} (2 x) - \frac{1}{4} cos^{2} (2 x) sin (2 x) - \frac{1}{4} x cos (4 x) + \frac{1}{16} sin (4 x) + \frac{1}{24} sin (6 x) - \frac{1}{8} sin (2 x) + C

解答ステップ

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

部分積分を適用する

= \frac{1}{2} cos^{2} (2 x) (x - \frac{1}{2} sin (2 x)) - \int - sin (4 x) (x - \frac{1}{2} sin (2 x)) d x

\int - sin (4 x) (x - \frac{1}{2} sin (2 x)) d x = \frac{1}{4} x cos (4 x) - \frac{1}{16} sin (4 x) - \frac{1}{24} sin (6 x) + \frac{1}{8} sin (2 x)

= \frac{1}{2} cos^{2} (2 x) (x - \frac{1}{2} sin (2 x)) - (\frac{1}{4} x cos (4 x) - \frac{1}{16} sin (4 x) - \frac{1}{24} sin (6 x) + \frac{1}{8} sin (2 x))

簡素化

\frac{1}{2} cos^{2} (2 x) (x - \frac{1}{2} sin (2 x)) - (\frac{1}{4} x cos (4 x) - \frac{1}{16} sin (4 x) - \frac{1}{24} sin (6 x) + \frac{1}{8} sin (2 x)) : \frac{1}{2} x cos^{2} (2 x) - \frac{1}{4} cos^{2} (2 x) sin (2 x) - \frac{1}{4} x cos (4 x) + \frac{1}{16} sin (4 x) + \frac{1}{24} sin (6 x) - \frac{1}{8} sin (2 x)

= \frac{1}{2} x cos^{2} (2 x) - \frac{1}{4} cos^{2} (2 x) sin (2 x) - \frac{1}{4} x cos (4 x) + \frac{1}{16} sin (4 x) + \frac{1}{24} sin (6 x) - \frac{1}{8} sin (2 x)

定数を解答に追加する

= \frac{1}{2} x cos^{2} (2 x) - \frac{1}{4} cos^{2} (2 x) sin (2 x) - \frac{1}{4} x cos (4 x) + \frac{1}{16} sin (4 x) + \frac{1}{24} sin (6 x) - \frac{1}{8} sin (2 x) + C