integral of (sec^6(x))/(tan^3(x))

Lösung

\int \frac{sec ^{6} ( x )}{tan ^{3} ( x )} d x

- \frac{1}{2} cot^{2} (x) + 2 ln ∣ tan (x) ∣ + \frac{1}{2} tan^{2} (x) + C

Schritte zur Lösung

\int \frac{sec ^{6} ( x )}{tan ^{3} ( x )} d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int \frac{1}{cos ^{3} ( x ) sin ^{3} ( x )} d x

Trigonometrische Substitution anwenden

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

Multipliziere aus

\frac{( 1 + u ^{2} ) ^{2}}{u ^{3}} : \frac{1}{u ^{3}} + \frac{2}{u} + u

= \int \frac{1}{u ^{3}} + \frac{2}{u} + u 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 \frac{1}{u ^{3}} d u + \int \frac{2}{u} d u + \int u d u

\int \frac{1}{u ^{3}} d u = - \frac{1}{2 u ^{2}}

\int \frac{2}{u} d u = 2 ln ∣ u ∣

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

= - \frac{1}{2 u ^{2}} + 2 ln ∣ u ∣ + \frac{u ^{2}}{2}

Setze in

u = tan (x)

ein

= - \frac{1}{2 tan ^{2} ( x )} + 2 ln ∣ tan (x) ∣ + \frac{tan ^{2} ( x )}{2}

Vereinfache

- \frac{1}{2 tan ^{2} ( x )} + 2 ln ∣ tan (x) ∣ + \frac{tan ^{2} ( x )}{2} : - \frac{1}{2} cot^{2} (x) + 2 ln ∣ tan (x) ∣ + \frac{1}{2} tan^{2} (x)

= - \frac{1}{2} cot^{2} (x) + 2 ln ∣ tan (x) ∣ + \frac{1}{2} tan^{2} (x)

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{2} cot^{2} (x) + 2 ln ∣ tan (x) ∣ + \frac{1}{2} tan^{2} (x) + C

x \to 0 - lim (e^{\frac{3}{x}})

x \to 0 lim (\frac{x + 5}{3 x})

d er i v a t i v e \frac{x ^{2}}{x - 9}

(x^{3} + 3 y) d x - x d y = 0

\int \frac{e ^{x}}{( e ^{x} - 9 ) ( e ^{2 x} + 1 )} d x