integral of (x-cos(x))/(1+sin(x))

解

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

x tan (x) + ln ∣ cos (x) ∣ - x sin (x) tan (x) - x cos (x) + ln ∣ tan (x) + sec (x) ∣ - ln ∣ 1 + sin (x) ∣ + C

解答ステップ

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

拡張

\frac{x - cos ( x )}{1 + sin ( x )} : \frac{x}{1 + sin ( x )} - \frac{cos ( x )}{1 + sin ( 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{x}{1 + sin ( x )} d x - \int \frac{cos ( x )}{1 + sin ( x )} d x

\int \frac{x}{1 + sin ( x )} d x = x tan (x) + ln ∣ cos (x) ∣ - x sin (x) tan (x) - x cos (x) + ln ∣ tan (x) + sec (x) ∣

\int \frac{cos ( x )}{1 + sin ( x )} d x = ln ∣ 1 + sin (x) ∣

= x tan (x) + ln ∣ cos (x) ∣ - x sin (x) tan (x) - x cos (x) + ln ∣ tan (x) + sec (x) ∣ - ln ∣ 1 + sin (x) ∣

定数を解答に追加する

= x tan (x) + ln ∣ cos (x) ∣ - x sin (x) tan (x) - x cos (x) + ln ∣ tan (x) + sec (x) ∣ - ln ∣ 1 + sin (x) ∣ + C