integral of tan^7(x)sec^5(x)

Lösung

\int tan^{7} (x) sec^{5} (x) d x

\frac{sec ^{11} ( x )}{11} - \frac{sec ^{9} ( x )}{3} + \frac{3 sec ^{7} ( x )}{7} - \frac{sec ^{5} ( x )}{5} + C

Schritte zur Lösung

\int tan^{7} (x) sec^{5} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int (- 1 + sec^{2} (x))^{3} tan (x) sec^{5} (x) d x

Wende U-Substitution an

= \int u^{4} (u^{2} - 1)^{3} d u

Multipliziere aus

u^{4} (u^{2} - 1)^{3} : u^{10} - 3 u^{8} + 3 u^{6} - u^{4}

= \int u^{10} - 3 u^{8} + 3 u^{6} - 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 u^{10} d u - \int 3 u^{8} d u + \int 3 u^{6} d u - \int u^{4} d u

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

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

\int 3 u^{6} d u = \frac{3 u ^{7}}{7}

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

= \frac{u ^{11}}{11} - \frac{u ^{9}}{3} + \frac{3 u ^{7}}{7} - \frac{u ^{5}}{5}

Setze in

u = sec (x)

ein

= \frac{sec ^{11} ( x )}{11} - \frac{sec ^{9} ( x )}{3} + \frac{3 sec ^{7} ( x )}{7} - \frac{sec ^{5} ( x )}{5}

Füge eine Konstante zur Lösung hinzu

= \frac{sec ^{11} ( x )}{11} - \frac{sec ^{9} ( x )}{3} + \frac{3 sec ^{7} ( x )}{7} - \frac{sec ^{5} ( x )}{5} + C

y^{^{''}} - 3 y^{^{'}} + 2 y = 4

\frac{\partial}{\partial x} (y - \frac{1}{x})

y^{^{'}} - y = 4 x

\frac{\partial}{\partial x} (arctan (x t))

d er i v a t i v e y = x^{2} + 2