integral of sec(θ)(sec(θ)+tan(θ))

解

\int sec (θ) (sec (θ) + tan (θ)) d θ

tan (θ) + sec (θ) + C

解答ステップ

\int sec (θ) (sec (θ) + tan (θ)) d θ

三角関数の公式を使用して書き換える

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

拡張

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

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

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

\int \frac{tan ( θ )}{cos ( θ )} d θ = sec (θ)

= tan (θ) + sec (θ)

定数を解答に追加する

= tan (θ) + sec (θ) + C