integral of sec(x)-tan(x)

解

\int sec (x) - tan (x) d x

ln ∣ tan (x) + sec (x) ∣ + ln ∣ cos (x) ∣ + C

解答ステップ

\int sec (x) - tan (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 sec (x) d x - \int tan (x) d x

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

\int tan (x) d x = - ln ∣ cos (x) ∣

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

簡素化

= ln ∣ tan (x) + sec (x) ∣ + ln ∣ cos (x) ∣

定数を解答に追加する

= ln ∣ tan (x) + sec (x) ∣ + ln ∣ cos (x) ∣ + C