integral of (sin^2(θ))/(cos^2(θ))

解

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

tan (θ) - θ + C

解答ステップ

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

三角関数の公式を使用して書き換える

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

拡張

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

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

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

\int 1 d θ = θ

= tan (θ) - θ

定数を解答に追加する

= tan (θ) - θ + C