integral of sec^5(θ/2)

Lösung

\int sec^{5} (\frac{θ}{2}) d θ

2 (\frac{sec ^{4} ( \frac{θ}{2} ) sin ( \frac{θ}{2} )}{4} + \frac{3}{4} (\frac{1}{2} sec (\frac{θ}{2}) tan (\frac{θ}{2}) + \frac{1}{2} ln tan (\frac{θ}{2}) + sec (\frac{θ}{2}))) + C

Schritte zur Lösung

\int sec^{5} (\frac{θ}{2}) d θ

Wende U-Substitution an

= \int sec^{5} (u) \cdot 2 d u

Entferne die Konstante:

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

= 2 \cdot \int sec^{5} (u) d u

Wende integrale Reduktion an

= 2 (\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) ∣

= 2 (\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 = \frac{θ}{2}

ein

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

Füge eine Konstante zur Lösung hinzu

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

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

\frac{\partial}{\partial x} (sin (5 x y + 8 yz + 9 x z))

\int \frac{1}{x ^{2} + π ^{2}} d x

\frac{\partial}{\partial x} (3^{s i n (3 x)})

\frac{d}{d x} ((4 x + 3)^{5})