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

Lösung

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

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

Schritte zur Lösung

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

Wende U-Substitution an

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

Entferne die Konstante:

\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

Multipliziere aus

\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

Wende die Summenregel an:

\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}}})

Setze in

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

ein

= \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}}}

Vereinfache

\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)

Füge eine Konstante zur Lösung hinzu

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

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