integral of sec^6(x)tan^4(x)

Lösung

\int sec^{6} (x) tan^{4} (x) d x

\frac{tan ^{5} ( x )}{5} + \frac{2 tan ^{7} ( x )}{7} + \frac{tan ^{9} ( x )}{9} + C

Schritte zur Lösung

\int sec^{6} (x) tan^{4} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int sec^{2} (x) (1 + tan^{2} (x))^{2} tan^{4} (x) d x

Wende U-Substitution an

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

Multipliziere aus

u^{4} (1 + u^{2})^{2} : u^{4} + 2 u^{6} + u^{8}

= \int u^{4} + 2 u^{6} + 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^{4} d u + \int 2 u^{6} d u + \int u^{8} d u

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

\int 2 u^{6} d u = \frac{2 u ^{7}}{7}

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

= \frac{u ^{5}}{5} + \frac{2 u ^{7}}{7} + \frac{u ^{9}}{9}

Setze in

u = tan (x)

ein

= \frac{tan ^{5} ( x )}{5} + \frac{2 tan ^{7} ( x )}{7} + \frac{tan ^{9} ( x )}{9}

Füge eine Konstante zur Lösung hinzu

= \frac{tan ^{5} ( x )}{5} + \frac{2 tan ^{7} ( x )}{7} + \frac{tan ^{9} ( x )}{9} + C

\int - x^{- 1} d x

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