integral of tan^7(x)sec^9(x)

Lösung

\int tan^{7} (x) sec^{9} (x) d x

\frac{sec ^{15} ( x )}{15} - \frac{3 sec ^{13} ( x )}{13} + \frac{3 sec ^{11} ( x )}{11} - \frac{sec ^{9} ( x )}{9} + C

Schritte zur Lösung

\int tan^{7} (x) sec^{9} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int (- 1 + sec^{2} (x))^{3} tan (x) sec^{9} (x) d x

Wende U-Substitution an

= \int u^{8} (u^{2} - 1)^{3} d u

Multipliziere aus

u^{8} (u^{2} - 1)^{3} : u^{14} - 3 u^{12} + 3 u^{10} - u^{8}

= \int u^{14} - 3 u^{12} + 3 u^{10} - u^{8} 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

= \int u^{14} d u - \int 3 u^{12} d u + \int 3 u^{10} d u - \int u^{8} d u

\int u^{14} d u = \frac{u ^{15}}{15}

\int 3 u^{12} d u = \frac{3 u ^{13}}{13}

\int 3 u^{10} d u = \frac{3 u ^{11}}{11}

\int u^{8} d u = \frac{u ^{9}}{9}

= \frac{u ^{15}}{15} - \frac{3 u ^{13}}{13} + \frac{3 u ^{11}}{11} - \frac{u ^{9}}{9}

Setze in

u = sec (x)

ein

= \frac{sec ^{15} ( x )}{15} - \frac{3 sec ^{13} ( x )}{13} + \frac{3 sec ^{11} ( x )}{11} - \frac{sec ^{9} ( x )}{9}

Füge eine Konstante zur Lösung hinzu

= \frac{sec ^{15} ( x )}{15} - \frac{3 sec ^{13} ( x )}{13} + \frac{3 sec ^{11} ( x )}{11} - \frac{sec ^{9} ( x )}{9} + C

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