integral of (cos(x)-tan(x))/(cos^2(x))

解

\int \frac{cos ( x ) - tan ( x )}{cos ^{2} ( x )} d x

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

解答ステップ

\int \frac{cos ( x ) - tan ( x )}{cos ^{2} ( x )} d x

拡張

\frac{cos ( x ) - tan ( x )}{cos ^{2} ( x )} : \frac{1}{cos ( x )} - \frac{tan ( x )}{cos ^{2} ( x )}

= \int \frac{1}{cos ( x )} - \frac{tan ( x )}{cos ^{2} ( 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}{cos ( x )} d x - \int \frac{tan ( x )}{cos ^{2} ( x )} d x

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

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

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

定数を解答に追加する

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