integral of (sec(x)-cot(x))^2

解

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

tan (x) - 2 ln tan (\frac{x}{2}) - x - cot (x) + C

解答ステップ

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

拡張

(sec (x) - cot (x))^{2} : sec^{2} (x) - 2 sec (x) cot (x) + cot^{2} (x)

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

\int sec^{2} (x) d x = tan (x)

\int 2 sec (x) cot (x) d x = 2 ln tan (\frac{x}{2})

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

= tan (x) - 2 ln tan (\frac{x}{2}) - x - cot (x)

定数を解答に追加する

= tan (x) - 2 ln tan (\frac{x}{2}) - x - cot (x) + C