integral of (csc(3x)-cot(3x))^2

解

\int (csc (3 x) - cot (3 x))^{2} d x

- x - \frac{2}{3} cot (3 x) + \frac{2}{3} csc (3 x) + C

解答ステップ

\int (csc (3 x) - cot (3 x))^{2} d x

拡張

(csc (3 x) - cot (3 x))^{2} : csc^{2} (3 x) - 2 csc (3 x) cot (3 x) + cot^{2} (3 x)

= \int csc^{2} (3 x) - 2 csc (3 x) cot (3 x) + cot^{2} (3 x) d x

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 csc^{2} (3 x) d x - \int 2 csc (3 x) cot (3 x) d x + \int cot^{2} (3 x) d x

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

\int 2 csc (3 x) cot (3 x) d x = - \frac{2}{3} csc (3 x)

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

= - \frac{1}{3} cot (3 x) - (- \frac{2}{3} csc (3 x)) - x - \frac{1}{3} cot (3 x)

簡素化

- \frac{1}{3} cot (3 x) - (- \frac{2}{3} csc (3 x)) - x - \frac{1}{3} cot (3 x) : - x - \frac{2}{3} cot (3 x) + \frac{2}{3} csc (3 x)

= - x - \frac{2}{3} cot (3 x) + \frac{2}{3} csc (3 x)

定数を解答に追加する

= - x - \frac{2}{3} cot (3 x) + \frac{2}{3} csc (3 x) + C