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

Lösung

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

\frac{sec ^{13} ( x )}{13} - \frac{3 sec ^{11} ( x )}{11} + \frac{sec ^{9} ( x )}{3} - \frac{sec ^{7} ( x )}{7} + C

Schritte zur Lösung

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Wende U-Substitution an

= \int u^{6} (u^{2} - 1)^{3} d u

Multipliziere aus

u^{6} (u^{2} - 1)^{3} : u^{12} - 3 u^{10} + 3 u^{8} - u^{6}

= \int u^{12} - 3 u^{10} + 3 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^{12} d u - \int 3 u^{10} d u + \int 3 u^{8} d u - \int u^{6} d u

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

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

\int 3 u^{8} d u = \frac{u ^{9}}{3}

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

= \frac{u ^{13}}{13} - \frac{3 u ^{11}}{11} + \frac{u ^{9}}{3} - \frac{u ^{7}}{7}

Setze in

u = sec (x)

ein

= \frac{sec ^{13} ( x )}{13} - \frac{3 sec ^{11} ( x )}{11} + \frac{sec ^{9} ( x )}{3} - \frac{sec ^{7} ( x )}{7}

Füge eine Konstante zur Lösung hinzu

= \frac{sec ^{13} ( x )}{13} - \frac{3 sec ^{11} ( x )}{11} + \frac{sec ^{9} ( x )}{3} - \frac{sec ^{7} ( x )}{7} + C

\int lo g_{5} (x) d x

\int ∣ 4 x + 1 ∣ d x

\int sin (e^{t}) d t

\frac{\partial}{\partial x} (\frac{4 x ^{2} y ^{2}}{x + y})

d er i v a t i v e f (x) = x^{2} - 6 x + 5