integral of (csc(x)+tan(x))^2

解

\int (csc (x) + tan (x))^{2} d x

- cot (x) + 2 ln ∣ tan (x) + sec (x) ∣ - x + tan (x) + C

解答ステップ

\int (csc (x) + tan (x))^{2} d x

拡張

(csc (x) + tan (x))^{2} : csc^{2} (x) + 2 csc (x) tan (x) + tan^{2} (x)

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

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

\int 2 csc (x) tan (x) d x = 2 ln ∣ tan (x) + sec (x) ∣

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

= - cot (x) + 2 ln ∣ tan (x) + sec (x) ∣ - x + tan (x)

定数を解答に追加する

= - cot (x) + 2 ln ∣ tan (x) + sec (x) ∣ - x + tan (x) + C