integral of 6/(sec(x)-1)

解

\int \frac{6}{sec ( x ) - 1} d x

6 (- cot (\frac{x}{2}) - x) + C

解答ステップ

\int \frac{6}{sec ( x ) - 1} d x

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 6 \cdot \int \frac{1}{sec ( x ) - 1} d x

\frac{1}{sec ( x ) - 1} = \frac{sec ( x )}{sec ( x ) - 1} - 1

= 6 \cdot \int \frac{sec ( x )}{sec ( x ) - 1} - 1 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

= 6 (\int \frac{sec ( x )}{sec ( x ) - 1} d x - \int 1 d x)

\int \frac{sec ( x )}{sec ( x ) - 1} d x = - cot (\frac{x}{2})

\int 1 d x = x

= 6 (- cot (\frac{x}{2}) - x)

定数を解答に追加する

= 6 (- cot (\frac{x}{2}) - x) + C