integral of cos^{(3)}(5x)sin^{-2}(5x)

解

\int cos^{(3)} (5 x) sin^{- 2} (5 x) d x

- \frac{1}{5} cos^{3} (5 x) cot (5 x) - \frac{3}{5} sin (5 x) + \frac{1}{5} sin^{3} (5 x) + C

解答ステップ

\int cos^{3} (5 x) sin^{- 2} (5 x) d x

部分積分を適用する

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

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

= - \frac{1}{5} cos^{3} (5 x) cot (5 x) - (\frac{3}{5} sin (5 x) - \frac{1}{5} sin^{3} (5 x))

簡素化

= - \frac{1}{5} cos^{3} (5 x) cot (5 x) - \frac{3}{5} sin (5 x) + \frac{1}{5} sin^{3} (5 x)

定数を解答に追加する

= - \frac{1}{5} cos^{3} (5 x) cot (5 x) - \frac{3}{5} sin (5 x) + \frac{1}{5} sin^{3} (5 x) + C