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Beliebt Rechnen >

integral of tan^5(θ)sec^7(θ)

Lösung

\int tan^{5} (θ) sec^{7} (θ) d θ

Lösung

\frac{sec ^{11} ( θ )}{11} - \frac{2 sec ^{9} ( θ )}{9} + \frac{sec ^{7} ( θ )}{7} + C

Schritte zur Lösung

\int tan^{5} (θ) sec^{7} (θ) d θ

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int (- 1 + sec^{2} (θ))^{2} tan (θ) sec^{7} (θ) d θ

Wende U-Substitution an

= \int u^{6} (u^{2} - 1)^{2} d u

Multipliziere aus

u^{6} (u^{2} - 1)^{2} : u^{10} - 2 u^{8} + u^{6}

= \int u^{10} - 2 u^{8} + u^{6} 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^{10} d u - \int 2 u^{8} d u + \int u^{6} d u

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

\int 2 u^{8} d u = \frac{2 u ^{9}}{9}

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

= \frac{u ^{11}}{11} - \frac{2 u ^{9}}{9} + \frac{u ^{7}}{7}

Setze in

u = sec (θ)

ein

= \frac{sec ^{11} ( θ )}{11} - \frac{2 sec ^{9} ( θ )}{9} + \frac{sec ^{7} ( θ )}{7}

Füge eine Konstante zur Lösung hinzu

= \frac{sec ^{11} ( θ )}{11} - \frac{2 sec ^{9} ( θ )}{9} + \frac{sec ^{7} ( θ )}{7} + C

Graph

Beliebte Beispiele

integral of 9/(sqrt(e^{9x)-7)}

\int \frac{9}{e ^{9 x} - 7} d x

integral of 1/((x^2+2x+3))

\int \frac{1}{( x ^{2} + 2 x + 3 )} d x

integral of 2/(sqrt(x^2+4))

\int \frac{2}{x ^{2} + 4} d x

integral of sec^2(x)e^{6tan(x)}

\int sec^{2} (x) e^{6 t a n (x)} d x

integral of (x-9)/(sqrt(x)+3)

\int \frac{x - 9}{x + 3} d x