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人気のある微分積分 >

integral of (12)/(sec(2x)-1)

解

\int \frac{12}{sec ( 2 x ) - 1} d x

解

6 (- 2 x - cot (x)) + C

解答ステップ

\int \frac{12}{sec ( 2 x ) - 1} d x

定数を除く:

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

= 12 \cdot \int \frac{1}{sec ( 2 x ) - 1} d x

u置換積分法を適用する

= 12 \cdot \int \frac{1}{2 ( sec ( u ) - 1 )} d u

定数を除く:

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

= 12 \cdot \frac{1}{2} \cdot \int \frac{1}{sec ( u ) - 1} d u

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

= 12 \cdot \frac{1}{2} \cdot \int \frac{sec ( u )}{sec ( u ) - 1} - 1 d u

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

= 12 \cdot \frac{1}{2} (\int \frac{sec ( u )}{sec ( u ) - 1} d u - \int 1 d u)

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

\int 1 d u = u

= 12 \cdot \frac{1}{2} (- cot (\frac{u}{2}) - u)

代用を戻す

u = 2 x

= 12 \cdot \frac{1}{2} (- cot (\frac{2 x}{2}) - 2 x)

簡素化

12 \cdot \frac{1}{2} (- cot (\frac{2 x}{2}) - 2 x) : 6 (- 2 x - cot (x))

= 6 (- 2 x - cot (x))

定数を解答に追加する

= 6 (- 2 x - cot (x)) + C

グラフ

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