integral of sec(x)(2sec(x)+5tan(x))

解

\int sec (x) (2 sec (x) + 5 tan (x)) d x

2 tan (x) + 5 sec (x) + C

解答ステップ

\int sec (x) (2 sec (x) + 5 tan (x)) d x

三角関数の公式を使用して書き換える

= \int \frac{2 + 5 tan ( x ) cos ( x )}{cos ^{2} ( x )} d x

拡張

\frac{2 + 5 tan ( x ) cos ( x )}{cos ^{2} ( x )} : \frac{2}{cos ^{2} ( x )} + \frac{5 tan ( x )}{cos ( x )}

= \int \frac{2}{cos ^{2} ( x )} + \frac{5 tan ( x )}{cos ( 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{2}{cos ^{2} ( x )} d x + \int \frac{5 tan ( x )}{cos ( x )} d x

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

\int \frac{5 tan ( x )}{cos ( x )} d x = 5 sec (x)

= 2 tan (x) + 5 sec (x)

定数を解答に追加する

= 2 tan (x) + 5 sec (x) + C