integral of (2+cot(x))/(cos(x))

解

\int \frac{2 + cot ( x )}{cos ( x )} d x

2 ln ∣ tan (x) + sec (x) ∣ + ln tan (\frac{x}{2}) + C

解答ステップ

\int \frac{2 + cot ( x )}{cos ( x )} d x

拡張

\frac{2 + cot ( x )}{cos ( x )} : \frac{2}{cos ( x )} + \frac{cot ( x )}{cos ( x )}

= \int \frac{2}{cos ( x )} + \frac{cot ( x )}{cos ( 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{2}{cos ( x )} d x + \int \frac{cot ( x )}{cos ( x )} d x

\int \frac{2}{cos ( x )} d x = 2 ln ∣ tan (x) + sec (x) ∣

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

= 2 ln ∣ tan (x) + sec (x) ∣ + ln tan (\frac{x}{2})

定数を解答に追加する

= 2 ln ∣ tan (x) + sec (x) ∣ + ln tan (\frac{x}{2}) + C