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Beliebt Rechnen >

integral of tan^7(x)sec^8(x)

Lösung

\int tan^{7} (x) sec^{8} (x) d x

Lösung

\frac{tan ^{8} ( x )}{8} + \frac{3 tan ^{10} ( x )}{10} + \frac{tan ^{12} ( x )}{4} + \frac{tan ^{14} ( x )}{14} + C

Schritte zur Lösung

\int tan^{7} (x) sec^{8} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int sec^{2} (x) (1 + tan^{2} (x))^{3} tan^{7} (x) d x

Wende U-Substitution an

= \int u^{7} (1 + u^{2})^{3} d u

Multipliziere aus

u^{7} (1 + u^{2})^{3} : u^{7} + 3 u^{9} + 3 u^{11} + u^{13}

= \int u^{7} + 3 u^{9} + 3 u^{11} + u^{13} 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^{7} d u + \int 3 u^{9} d u + \int 3 u^{11} d u + \int u^{13} d u

\int u^{7} d u = \frac{u ^{8}}{8}

\int 3 u^{9} d u = \frac{3 u ^{10}}{10}

\int 3 u^{11} d u = \frac{u ^{12}}{4}

\int u^{13} d u = \frac{u ^{14}}{14}

= \frac{u ^{8}}{8} + \frac{3 u ^{10}}{10} + \frac{u ^{12}}{4} + \frac{u ^{14}}{14}

Setze in

u = tan (x)

ein

= \frac{tan ^{8} ( x )}{8} + \frac{3 tan ^{10} ( x )}{10} + \frac{tan ^{12} ( x )}{4} + \frac{tan ^{14} ( x )}{14}

Füge eine Konstante zur Lösung hinzu

= \frac{tan ^{8} ( x )}{8} + \frac{3 tan ^{10} ( x )}{10} + \frac{tan ^{12} ( x )}{4} + \frac{tan ^{14} ( x )}{14} + C

Graph

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