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人気のある微分積分 >

integral of cot^{(-5)/4}(2x)csc^4(2x)

解

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

解

\frac{2}{7} 4 tan (2 x) (- cot^{\frac{7}{4}} (2 x) 4 cot (2 x) + 7) + C

解答ステップ

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

u置換積分法を適用する

= \int \frac{2 ( u ^{\frac{8}{5}} + 1 )}{5 u ^{\frac{12}{5}}} d u

定数を除く:

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

= \frac{2}{5} \cdot \int \frac{u ^{\frac{8}{5}} + 1}{u ^{\frac{12}{5}}} d u

拡張

\frac{u ^{\frac{8}{5}} + 1}{u ^{\frac{12}{5}}} : \frac{1}{u ^{\frac{4}{5}}} + \frac{1}{u ^{\frac{12}{5}}}

= \frac{2}{5} \cdot \int \frac{1}{u ^{\frac{4}{5}}} + \frac{1}{u ^{\frac{12}{5}}} 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

= \frac{2}{5} (\int \frac{1}{u ^{\frac{4}{5}}} d u + \int \frac{1}{u ^{\frac{12}{5}}} d u)

\int \frac{1}{u ^{\frac{4}{5}}} d u = 5 u^{\frac{1}{5}}

\int \frac{1}{u ^{\frac{12}{5}}} d u = - \frac{5}{7 u ^{\frac{7}{5}}}

= \frac{2}{5} (5 u^{\frac{1}{5}} - \frac{5}{7 u ^{\frac{7}{5}}})

代用を戻す

u = cot^{\frac{- 5}{4}} (2 x)

= \frac{2}{5} 5 (cot^{\frac{- 5}{4}} (2 x))^{\frac{1}{5}} - \frac{5}{7 ( cot ^{\frac{- 5}{4}} ( 2 x ) ) ^{\frac{7}{5}}}

簡素化

\frac{2}{5} 5 (cot^{\frac{- 5}{4}} (2 x))^{\frac{1}{5}} - \frac{5}{7 ( cot ^{\frac{- 5}{4}} ( 2 x ) ) ^{\frac{7}{5}}} : \frac{2}{7} 4 tan (2 x) (- cot^{\frac{7}{4}} (2 x) 4 cot (2 x) + 7)

= \frac{2}{7} 4 tan (2 x) (- cot^{\frac{7}{4}} (2 x) 4 cot (2 x) + 7)

定数を解答に追加する

= \frac{2}{7} 4 tan (2 x) (- cot^{\frac{7}{4}} (2 x) 4 cot (2 x) + 7) + C

グラフ

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