integral of (1-(x+2)cot(x))/(sin(x))

解

\int \frac{1 - ( x + 2 ) cot ( x )}{sin ( x )} d x

x csc (x) + 2 csc (x) + C

解答ステップ

\int \frac{1 - ( x + 2 ) cot ( x )}{sin ( x )} d x

拡張

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

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

\int \frac{1}{sin ( x )} d x = ln tan (\frac{x}{2})

\int \frac{x cot ( x )}{sin ( x )} d x = - x csc (x) + ln tan (\frac{x}{2})

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

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

簡素化

ln tan (\frac{x}{2}) - (- x csc (x) + ln tan (\frac{x}{2})) - (- 2 csc (x)) : x csc (x) + 2 csc (x)

= x csc (x) + 2 csc (x)

定数を解答に追加する

= x csc (x) + 2 csc (x) + C