integral of 3sec^6(x)

Lösung

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

3 tan (x) + 2 tan^{3} (x) + \frac{3}{5} tan^{5} (x) + C

Schritte zur Lösung

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

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \int sec^{6} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Wende U-Substitution an

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

Multipliziere aus

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

= 3 \cdot \int 1 + 2 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

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

\int 1 d u = u

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

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

= 3 (u + \frac{2 u ^{3}}{3} + \frac{u ^{5}}{5})

Setze in

u = tan (x)

ein

= 3 (tan (x) + \frac{2 tan ^{3} ( x )}{3} + \frac{tan ^{5} ( x )}{5})

Vereinfache

3 (tan (x) + \frac{2 tan ^{3} ( x )}{3} + \frac{tan ^{5} ( x )}{5}) : 3 tan (x) + 2 tan^{3} (x) + \frac{3}{5} tan^{5} (x)

= 3 tan (x) + 2 tan^{3} (x) + \frac{3}{5} tan^{5} (x)

Füge eine Konstante zur Lösung hinzu

= 3 tan (x) + 2 tan^{3} (x) + \frac{3}{5} tan^{5} (x) + C

\frac{\partial}{\partial y} (\frac{8 x}{4 x ^{2} + 5 y ^{2}})

a re a 5 x, 2 x^{2}, 0, 2.5

\int x^{2} cos^{5} (2 x) d x

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

\int e^{x} 4 + 3 e^{x} d x