integral of (sin(3x))/(csc^2(4x))

解

\int \frac{sin ( 3 x )}{csc ^{2} ( 4 x )} d x

- \frac{1}{3} sin^{2} (4 x) cos (3 x) + \frac{2}{165} (- 5 cos (11 x) - 11 cos (5 x)) + C

解答ステップ

\int \frac{sin ( 3 x )}{csc ^{2} ( 4 x )} d x

部分積分を適用する

= - \frac{1}{3} sin^{2} (4 x) cos (3 x) - \int - \frac{4}{3} sin (8 x) cos (3 x) d x

\int - \frac{4}{3} sin (8 x) cos (3 x) d x = - \frac{2}{165} (- 5 cos (11 x) - 11 cos (5 x))

= - \frac{1}{3} sin^{2} (4 x) cos (3 x) - (- \frac{2}{165} (- 5 cos (11 x) - 11 cos (5 x)))

簡素化

= - \frac{1}{3} sin^{2} (4 x) cos (3 x) + \frac{2}{165} (- 5 cos (11 x) - 11 cos (5 x))

定数を解答に追加する

= - \frac{1}{3} sin^{2} (4 x) cos (3 x) + \frac{2}{165} (- 5 cos (11 x) - 11 cos (5 x)) + C