integral of (cot(x)-1)^2

解

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

- cot (x) - 2 ln ∣ sin (x) ∣ + C

解答ステップ

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

拡張

(cot (x) - 1)^{2} : cot^{2} (x) - 2 cot (x) + 1

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

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

\int 2 cot (x) d x = 2 ln ∣ sin (x) ∣

\int 1 d x = x

= - x - cot (x) - 2 ln ∣ sin (x) ∣ + x

簡素化

= - cot (x) - 2 ln ∣ sin (x) ∣

定数を解答に追加する

= - cot (x) - 2 ln ∣ sin (x) ∣ + C