integral of (sin(2u)-cos(3u))^{(2)}

解

\int (sin (2 u) - cos (3 u))^{(2)} d u

u - \frac{1}{8} sin (4 u) + \frac{1}{5} cos (5 u) + \frac{1}{12} sin (6 u) - cos (u) + C

解答ステップ

\int (sin (2 u) - cos (3 u))^{2} d u

因数

sin (2 u) - cos (3 u) : - (- sin (2 u) + cos (3 u))

= \int (- (- sin (2 u) + cos (3 u)))^{2} d u

簡素化

(- (- sin (2 u) + cos (3 u)))^{2} : (- sin (2 u) + cos (3 u))^{2}

= \int (- sin (2 u) + cos (3 u))^{2} d u

拡張

(- sin (2 u) + cos (3 u))^{2} : sin^{2} (2 u) - 2 sin (2 u) cos (3 u) + cos^{2} (3 u)

= \int sin^{2} (2 u) - 2 sin (2 u) cos (3 u) + cos^{2} (3 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

= \int sin^{2} (2 u) d u - \int 2 sin (2 u) cos (3 u) d u + \int cos^{2} (3 u) d u

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

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

\int cos^{2} (3 u) d u = \frac{1}{2} (u + \frac{1}{6} sin (6 u))

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

簡素化

\frac{1}{2} (u - \frac{1}{4} sin (4 u)) - (- \frac{1}{5} cos (5 u) + cos (u)) + \frac{1}{2} (u + \frac{1}{6} sin (6 u)) : u - \frac{1}{8} sin (4 u) + \frac{1}{5} cos (5 u) + \frac{1}{12} sin (6 u) - cos (u)

= u - \frac{1}{8} sin (4 u) + \frac{1}{5} cos (5 u) + \frac{1}{12} sin (6 u) - cos (u)

定数を解答に追加する

= u - \frac{1}{8} sin (4 u) + \frac{1}{5} cos (5 u) + \frac{1}{12} sin (6 u) - cos (u) + C