integral of cot^{10}(x)csc^4(x)

Lösung

\int cot^{10} (x) csc^{4} (x) d x

\frac{1}{8192} (cot^{13} (\frac{x}{2}) sec^{6} (\frac{x}{2}) (- tan^{2} (\frac{x}{2}) + 1)^{10} - 2 (\frac{6 tan ^{13} ( \frac{x}{2} )}{13} - \frac{35 tan ^{11} ( \frac{x}{2} )}{11} + 8 tan^{9} (\frac{x}{2}) - 6 tan^{7} (\frac{x}{2}) - 10 tan^{5} (\frac{x}{2}) + 21 tan^{3} (\frac{x}{2}) - 36 cot (\frac{x}{2}) + 42 cot^{3} (\frac{x}{2}) - 15 cot^{5} (\frac{x}{2}) - 8 cot^{7} (\frac{x}{2}) + 10 cot^{9} (\frac{x}{2}) - \frac{42 cot ^{11} ( \frac{x}{2} )}{11} + \frac{7 cot ^{13} ( \frac{x}{2} )}{13})) + C

Schritte zur Lösung

\int cot^{10} (x) csc^{4} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int \frac{cos ^{10} ( x )}{sin ^{14} ( x )} d x

Wende U-Substitution an

= \int \frac{( - u ^{2} + 1 ) ^{10} ( u ^{2} + 1 ) ^{3}}{8192 u ^{14}} d u

Entferne die Konstante:

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

= \frac{1}{8192} \cdot \int \frac{( - u ^{2} + 1 ) ^{10} ( u ^{2} + 1 ) ^{3}}{u ^{14}} d u

Wende die partielle Integration an

= \frac{1}{8192} (\frac{( - u ^{2} + 1 ) ^{10} ( u ^{2} + 1 ) ^{3}}{u ^{13}} - \int \frac{2 ( - u ^{2} + 1 ) ^{9} ( u ^{2} + 1 ) ^{2} ( - 6 u ^{4} - 7 u ^{2} - 7 )}{u ^{14}} d u)

\int \frac{2 ( - u ^{2} + 1 ) ^{9} ( u ^{2} + 1 ) ^{2} ( - 6 u ^{4} - 7 u ^{2} - 7 )}{u ^{14}} d u = 2 (\frac{6 u ^{13}}{13} - \frac{35 u ^{11}}{11} + 8 u^{9} - 6 u^{7} - 10 u^{5} + 21 u^{3} - \frac{36}{u} + \frac{42}{u ^{3}} - \frac{15}{u ^{5}} - \frac{8}{u ^{7}} + \frac{10}{u ^{9}} - \frac{42}{11 u ^{11}} + \frac{7}{13 u ^{13}})

= \frac{1}{8192} (\frac{( - u ^{2} + 1 ) ^{10} ( u ^{2} + 1 ) ^{3}}{u ^{13}} - 2 (\frac{6 u ^{13}}{13} - \frac{35 u ^{11}}{11} + 8 u^{9} - 6 u^{7} - 10 u^{5} + 21 u^{3} - \frac{36}{u} + \frac{42}{u ^{3}} - \frac{15}{u ^{5}} - \frac{8}{u ^{7}} + \frac{10}{u ^{9}} - \frac{42}{11 u ^{11}} + \frac{7}{13 u ^{13}}))

Setze in

u = tan (\frac{x}{2})

ein

= \frac{1}{8192} (\frac{( - tan ^{2} ( \frac{x}{2} ) + 1 ) ^{10} ( tan ^{2} ( \frac{x}{2} ) + 1 ) ^{3}}{tan ^{13} ( \frac{x}{2} )} - 2 (\frac{6 tan ^{13} ( \frac{x}{2} )}{13} - \frac{35 tan ^{11} ( \frac{x}{2} )}{11} + 8 tan^{9} (\frac{x}{2}) - 6 tan^{7} (\frac{x}{2}) - 10 tan^{5} (\frac{x}{2}) + 21 tan^{3} (\frac{x}{2}) - \frac{36}{tan ( \frac{x}{2} )} + \frac{42}{tan ^{3} ( \frac{x}{2} )} - \frac{15}{tan ^{5} ( \frac{x}{2} )} - \frac{8}{tan ^{7} ( \frac{x}{2} )} + \frac{10}{tan ^{9} ( \frac{x}{2} )} - \frac{42}{11 tan ^{11} ( \frac{x}{2} )} + \frac{7}{13 tan ^{13} ( \frac{x}{2} )}))

Vereinfache

\frac{1}{8192} (\frac{( - tan ^{2} ( \frac{x}{2} ) + 1 ) ^{10} ( tan ^{2} ( \frac{x}{2} ) + 1 ) ^{3}}{tan ^{13} ( \frac{x}{2} )} - 2 (\frac{6 tan ^{13} ( \frac{x}{2} )}{13} - \frac{35 tan ^{11} ( \frac{x}{2} )}{11} + 8 tan^{9} (\frac{x}{2}) - 6 tan^{7} (\frac{x}{2}) - 10 tan^{5} (\frac{x}{2}) + 21 tan^{3} (\frac{x}{2}) - \frac{36}{tan ( \frac{x}{2} )} + \frac{42}{tan ^{3} ( \frac{x}{2} )} - \frac{15}{tan ^{5} ( \frac{x}{2} )} - \frac{8}{tan ^{7} ( \frac{x}{2} )} + \frac{10}{tan ^{9} ( \frac{x}{2} )} - \frac{42}{11 tan ^{11} ( \frac{x}{2} )} + \frac{7}{13 tan ^{13} ( \frac{x}{2} )})) : \frac{1}{8192} (cot^{13} (\frac{x}{2}) sec^{6} (\frac{x}{2}) (- tan^{2} (\frac{x}{2}) + 1)^{10} - 2 (\frac{6 tan ^{13} ( \frac{x}{2} )}{13} - \frac{35 tan ^{11} ( \frac{x}{2} )}{11} + 8 tan^{9} (\frac{x}{2}) - 6 tan^{7} (\frac{x}{2}) - 10 tan^{5} (\frac{x}{2}) + 21 tan^{3} (\frac{x}{2}) - 36 cot (\frac{x}{2}) + 42 cot^{3} (\frac{x}{2}) - 15 cot^{5} (\frac{x}{2}) - 8 cot^{7} (\frac{x}{2}) + 10 cot^{9} (\frac{x}{2}) - \frac{42 cot ^{11} ( \frac{x}{2} )}{11} + \frac{7 cot ^{13} ( \frac{x}{2} )}{13}))

= \frac{1}{8192} (cot^{13} (\frac{x}{2}) sec^{6} (\frac{x}{2}) (- tan^{2} (\frac{x}{2}) + 1)^{10} - 2 (\frac{6 tan ^{13} ( \frac{x}{2} )}{13} - \frac{35 tan ^{11} ( \frac{x}{2} )}{11} + 8 tan^{9} (\frac{x}{2}) - 6 tan^{7} (\frac{x}{2}) - 10 tan^{5} (\frac{x}{2}) + 21 tan^{3} (\frac{x}{2}) - 36 cot (\frac{x}{2}) + 42 cot^{3} (\frac{x}{2}) - 15 cot^{5} (\frac{x}{2}) - 8 cot^{7} (\frac{x}{2}) + 10 cot^{9} (\frac{x}{2}) - \frac{42 cot ^{11} ( \frac{x}{2} )}{11} + \frac{7 cot ^{13} ( \frac{x}{2} )}{13}))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{8192} (cot^{13} (\frac{x}{2}) sec^{6} (\frac{x}{2}) (- tan^{2} (\frac{x}{2}) + 1)^{10} - 2 (\frac{6 tan ^{13} ( \frac{x}{2} )}{13} - \frac{35 tan ^{11} ( \frac{x}{2} )}{11} + 8 tan^{9} (\frac{x}{2}) - 6 tan^{7} (\frac{x}{2}) - 10 tan^{5} (\frac{x}{2}) + 21 tan^{3} (\frac{x}{2}) - 36 cot (\frac{x}{2}) + 42 cot^{3} (\frac{x}{2}) - 15 cot^{5} (\frac{x}{2}) - 8 cot^{7} (\frac{x}{2}) + 10 cot^{9} (\frac{x}{2}) - \frac{42 cot ^{11} ( \frac{x}{2} )}{11} + \frac{7 cot ^{13} ( \frac{x}{2} )}{13})) + C

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