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

解

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

- \frac{1}{10} cos^{2} (2 x) sin (5 x) + \frac{1}{6} cos^{2} (2 x) sin (3 x) + \frac{1}{15} sin (x) + \frac{1}{90} sin (9 x) - \frac{1}{42} sin (7 x) + C

解答ステップ

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

部分積分を適用する

= \frac{1}{2} cos^{2} (2 x) (- \frac{1}{5} sin (5 x) + \frac{1}{3} sin (3 x)) - \int - sin (4 x) (- \frac{1}{5} sin (5 x) + \frac{1}{3} sin (3 x)) d x

\int - sin (4 x) (- \frac{1}{5} sin (5 x) + \frac{1}{3} sin (3 x)) d x = - \frac{1}{15} sin (x) - \frac{1}{90} sin (9 x) + \frac{1}{42} sin (7 x)

= \frac{1}{2} cos^{2} (2 x) (- \frac{1}{5} sin (5 x) + \frac{1}{3} sin (3 x)) - (- \frac{1}{15} sin (x) - \frac{1}{90} sin (9 x) + \frac{1}{42} sin (7 x))

簡素化

\frac{1}{2} cos^{2} (2 x) (- \frac{1}{5} sin (5 x) + \frac{1}{3} sin (3 x)) - (- \frac{1}{15} sin (x) - \frac{1}{90} sin (9 x) + \frac{1}{42} sin (7 x)) : - \frac{1}{10} cos^{2} (2 x) sin (5 x) + \frac{1}{6} cos^{2} (2 x) sin (3 x) + \frac{1}{15} sin (x) + \frac{1}{90} sin (9 x) - \frac{1}{42} sin (7 x)

= - \frac{1}{10} cos^{2} (2 x) sin (5 x) + \frac{1}{6} cos^{2} (2 x) sin (3 x) + \frac{1}{15} sin (x) + \frac{1}{90} sin (9 x) - \frac{1}{42} sin (7 x)

定数を解答に追加する

= - \frac{1}{10} cos^{2} (2 x) sin (5 x) + \frac{1}{6} cos^{2} (2 x) sin (3 x) + \frac{1}{15} sin (x) + \frac{1}{90} sin (9 x) - \frac{1}{42} sin (7 x) + C