integral of ((e^{tan(x)}))/(cos^2(x))

解

\int \frac{( e ^{t a n (x)} )}{cos ^{2} ( x )} d x

e^{t a n (x)} + C

解答ステップ

\int \frac{e ^{t a n (x)}}{cos ^{2} ( x )} d x

三角関数の公式を使用して書き換える

= \int \frac{e ^{t a n (x)}}{1 - sin ^{2} ( x )} d x

部分積分を適用する

= e^{t a n (x)} tan (x) - \int e^{t a n (x)} sec^{2} (x) tan (x) d x

\int e^{t a n (x)} sec^{2} (x) tan (x) d x = tan (x) e^{t a n (x)} - e^{t a n (x)}

= e^{t a n (x)} tan (x) - (tan (x) e^{t a n (x)} - e^{t a n (x)})

簡素化

e^{t a n (x)} tan (x) - (tan (x) e^{t a n (x)} - e^{t a n (x)}) : e^{t a n (x)}

= e^{t a n (x)}

定数を解答に追加する

= e^{t a n (x)} + C