integral of sec^4(x)tan^2(x)

Lösung

\int sec^{4} (x) tan^{2} (x) d x

\frac{tan ^{3} ( x )}{3} + \frac{tan ^{5} ( x )}{5} + C

Schritte zur Lösung

\int sec^{4} (x) tan^{2} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Wende U-Substitution an

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

Multipliziere aus

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

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

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

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

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

Setze in

u = tan (x)

ein

= \frac{tan ^{3} ( x )}{3} + \frac{tan ^{5} ( x )}{5}

Füge eine Konstante zur Lösung hinzu

= \frac{tan ^{3} ( x )}{3} + \frac{tan ^{5} ( x )}{5} + C

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

(x^{2} - 1) \frac{d y}{d x} + 2 y = (x + 1)^{2}

d er i v a t i v e f (x) = (2 x + 3)^{2}

\int_{0}^{2} 2 x^{3} x^{2} + 4 d x

x \to 1 lim (x^{3} - 3)