integral of ((cos(θ)-sin(θ)))/(cos^2(θ))

解

\int \frac{( cos ( θ ) - sin ( θ ) )}{cos ^{2} ( θ )} d θ

ln ∣ tan (θ) + sec (θ) ∣ - sec (θ) + C

解答ステップ

\int \frac{cos ( θ ) - sin ( θ )}{cos ^{2} ( θ )} d θ

拡張

\frac{cos ( θ ) - sin ( θ )}{cos ^{2} ( θ )} : \frac{1}{cos ( θ )} - \frac{sin ( θ )}{cos ^{2} ( θ )}

= \int \frac{1}{cos ( θ )} - \frac{sin ( θ )}{cos ^{2} ( θ )} d θ

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 ( θ )} d θ - \int \frac{sin ( θ )}{cos ^{2} ( θ )} d θ

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

\int \frac{sin ( θ )}{cos ^{2} ( θ )} d θ = sec (θ)

= ln ∣ tan (θ) + sec (θ) ∣ - sec (θ)

定数を解答に追加する

= ln ∣ tan (θ) + sec (θ) ∣ - sec (θ) + C