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Beliebt Rechnen >

integral of sec^5(3x)

Lösung

\int sec^{5} (3 x) d x

Lösung

\frac{1}{12} sec^{4} (3 x) sin (3 x) + \frac{1}{8} (sec (3 x) tan (3 x) + ln ∣ tan (3 x) + sec (3 x) ∣) + C

Schritte zur Lösung

\int sec^{5} (3 x) d x

Wende U-Substitution an

= \int sec^{5} (u) \frac{1}{3} d u

Entferne die Konstante:

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

= \frac{1}{3} \cdot \int sec^{5} (u) d u

Wende integrale Reduktion an

= \frac{1}{3} (\frac{sec ^{4} ( u ) sin ( u )}{4} + \frac{3}{4} \cdot \int sec^{3} (u) d u)

\int sec^{3} (u) d u = \frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣

= \frac{1}{3} (\frac{sec ^{4} ( u ) sin ( u )}{4} + \frac{3}{4} (\frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣))

Setze in

u = 3 x

ein

= \frac{1}{3} (\frac{sec ^{4} ( 3 x ) sin ( 3 x )}{4} + \frac{3}{4} (\frac{1}{2} sec (3 x) tan (3 x) + \frac{1}{2} ln ∣ tan (3 x) + sec (3 x) ∣))

Vereinfache

\frac{1}{3} (\frac{sec ^{4} ( 3 x ) sin ( 3 x )}{4} + \frac{3}{4} (\frac{1}{2} sec (3 x) tan (3 x) + \frac{1}{2} ln ∣ tan (3 x) + sec (3 x) ∣)) : \frac{1}{12} sec^{4} (3 x) sin (3 x) + \frac{1}{8} (sec (3 x) tan (3 x) + ln ∣ tan (3 x) + sec (3 x) ∣)

= \frac{1}{12} sec^{4} (3 x) sin (3 x) + \frac{1}{8} (sec (3 x) tan (3 x) + ln ∣ tan (3 x) + sec (3 x) ∣)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{12} sec^{4} (3 x) sin (3 x) + \frac{1}{8} (sec (3 x) tan (3 x) + ln ∣ tan (3 x) + sec (3 x) ∣) + C

Graph

Beliebte Beispiele

derivative of (2xy/((2-x^2)))

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

derivative cos(6y)

d er i v a t i v e cos (6 y)

derivative of (ax+b/(cx+d))

\frac{d}{d x} (\frac{a x + b}{c x + d})

limit as x approaches 1 of arccos(x)

x \to 1 lim (arccos (x))

d/(ds)(6/(s^2+36))

\frac{d}{d s} (\frac{6}{s ^{2} + 36})