integral of (7sin(x)+2sec(x))/(tan(x))

解

\int \frac{7 sin ( x ) + 2 sec ( x )}{tan ( x )} d x

7 sin (x) + 2 ln tan (\frac{x}{2}) + C

解答ステップ

\int \frac{7 sin ( x ) + 2 sec ( x )}{tan ( x )} d x

拡張

\frac{7 sin ( x ) + 2 sec ( x )}{tan ( x )} : \frac{7 sin ( x )}{tan ( x )} + \frac{2 sec ( x )}{tan ( x )}

= \int \frac{7 sin ( x )}{tan ( x )} + \frac{2 sec ( x )}{tan ( 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{7 sin ( x )}{tan ( x )} d x + \int \frac{2 sec ( x )}{tan ( x )} d x

\int \frac{7 sin ( x )}{tan ( x )} d x = 7 sin (x)

\int \frac{2 sec ( x )}{tan ( x )} d x = 2 ln tan (\frac{x}{2})

= 7 sin (x) + 2 ln tan (\frac{x}{2})

定数を解答に追加する

= 7 sin (x) + 2 ln tan (\frac{x}{2}) + C