integral of tan^5(x)sec(x)

Lösung

\int tan^{5} (x) sec (x) d x

sec (x) - \frac{2 sec ^{3} ( x )}{3} + \frac{sec ^{5} ( x )}{5} + C

Schritte zur Lösung

\int tan^{5} (x) sec (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Wende U-Substitution an

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

Multipliziere aus

(- 1 + u^{2})^{2} : 1 - 2 u^{2} + u^{4}

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

\int 1 d u = u

\int 2 u^{2} d u = \frac{2 u ^{3}}{3}

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

= u - \frac{2 u ^{3}}{3} + \frac{u ^{5}}{5}

Setze in

u = sec (x)

ein

= sec (x) - \frac{2 sec ^{3} ( x )}{3} + \frac{sec ^{5} ( x )}{5}

Füge eine Konstante zur Lösung hinzu

= sec (x) - \frac{2 sec ^{3} ( x )}{3} + \frac{sec ^{5} ( x )}{5} + C

ma c l a u r in 5 e^{- x}

\frac{d}{d x} ((sin (x))^{l n (x)})

\frac{\partial}{\partial x} (x^{2} y + 5 y^{3})

\frac{d x}{d t} = 3 t^{2} (x^{2} + 4)

\frac{\partial}{\partial x} (9 \frac{x yz}{x + y + z})