integral of 4/5 csc^2(x/5)

解

\int \frac{4}{5} csc^{2} (\frac{x}{5}) d x

- 4 cot (\frac{x}{5}) + C

解答ステップ

\int \frac{4}{5} csc^{2} (\frac{x}{5}) d x

定数を除く:

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

= \frac{4}{5} \cdot \int csc^{2} (\frac{x}{5}) d x

三角関数の公式を使用して書き換える

= \frac{4}{5} \cdot \int 1 + cot^{2} (\frac{x}{5}) 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

= \frac{4}{5} (\int 1 d x + \int cot^{2} (\frac{x}{5}) d x)

\int 1 d x = x

\int cot^{2} (\frac{x}{5}) d x = - x - 5 cot (\frac{x}{5})

= \frac{4}{5} (x - x - 5 cot (\frac{x}{5}))

簡素化

\frac{4}{5} (x - x - 5 cot (\frac{x}{5})) : - 4 cot (\frac{x}{5})

= - 4 cot (\frac{x}{5})

定数を解答に追加する

= - 4 cot (\frac{x}{5}) + C