integral of sin^3(x)sin(3x)

解

\int sin^{3} (x) sin (3 x) d x

- sin (3 x) cos (x) + \frac{1}{3} cos^{3} (x) sin (3 x) - \frac{1}{8} x + \frac{13}{32} sin (4 x) + \frac{23}{32} sin (2 x) - \frac{1}{8} cos^{2} (x) sin (4 x) - \frac{1}{4} cos^{2} (x) sin (2 x) + \frac{1}{96} sin (6 x) + C

解答ステップ

\int sin^{3} (x) sin (3 x) d x

部分積分を適用する

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

\int - 3 cos (3 x) cos (x) + cos^{3} (x) cos (3 x) d x = \frac{1}{8} x - \frac{13}{32} sin (4 x) - \frac{23}{32} sin (2 x) + \frac{1}{8} cos^{2} (x) sin (4 x) + \frac{1}{4} cos^{2} (x) sin (2 x) - \frac{1}{96} sin (6 x)

= sin (3 x) (- cos (x) + \frac{1}{3} cos^{3} (x)) - (\frac{1}{8} x - \frac{13}{32} sin (4 x) - \frac{23}{32} sin (2 x) + \frac{1}{8} cos^{2} (x) sin (4 x) + \frac{1}{4} cos^{2} (x) sin (2 x) - \frac{1}{96} sin (6 x))

簡素化

sin (3 x) (- cos (x) + \frac{1}{3} cos^{3} (x)) - (\frac{1}{8} x - \frac{13}{32} sin (4 x) - \frac{23}{32} sin (2 x) + \frac{1}{8} cos^{2} (x) sin (4 x) + \frac{1}{4} cos^{2} (x) sin (2 x) - \frac{1}{96} sin (6 x)) : - sin (3 x) cos (x) + \frac{1}{3} cos^{3} (x) sin (3 x) - \frac{1}{8} x + \frac{13}{32} sin (4 x) + \frac{23}{32} sin (2 x) - \frac{1}{8} cos^{2} (x) sin (4 x) - \frac{1}{4} cos^{2} (x) sin (2 x) + \frac{1}{96} sin (6 x)

= - sin (3 x) cos (x) + \frac{1}{3} cos^{3} (x) sin (3 x) - \frac{1}{8} x + \frac{13}{32} sin (4 x) + \frac{23}{32} sin (2 x) - \frac{1}{8} cos^{2} (x) sin (4 x) - \frac{1}{4} cos^{2} (x) sin (2 x) + \frac{1}{96} sin (6 x)

定数を解答に追加する

= - sin (3 x) cos (x) + \frac{1}{3} cos^{3} (x) sin (3 x) - \frac{1}{8} x + \frac{13}{32} sin (4 x) + \frac{23}{32} sin (2 x) - \frac{1}{8} cos^{2} (x) sin (4 x) - \frac{1}{4} cos^{2} (x) sin (2 x) + \frac{1}{96} sin (6 x) + C