integral of (12+csc(x)(sin(x)+csc(x)))

解

\int (12 + csc (x) (sin (x) + csc (x))) d x

- cot (x) + 13 x + C

解答ステップ

\int 12 + csc (x) (sin (x) + csc (x)) d x

拡張

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

= \int 12 + csc (x) sin (x) + csc^{2} (x) d x

簡素化

12 + csc (x) sin (x) + csc^{2} (x) : csc^{2} (x) + 13

= \int csc^{2} (x) + 13 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 13 d x

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

\int 13 d x = 13 x

= - cot (x) + 13 x

定数を解答に追加する

= - cot (x) + 13 x + C