integral of 5sec^5(x)

Lösung

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

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

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Wende integrale Reduktion an

= 5 (\frac{sec ^{4} ( x ) sin ( x )}{4} + \frac{3}{4} \cdot \int sec^{3} (x) d x)

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

= 5 (\frac{sec ^{4} ( x ) sin ( x )}{4} + \frac{3}{4} (\frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣))

Füge eine Konstante zur Lösung hinzu

= 5 (\frac{sec ^{4} ( x ) sin ( x )}{4} + \frac{3}{4} (\frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣)) + C

\int - e^{- t} + 1 d t

t an g e n t f (x) = x^{4} - 10 x^{2} + 9, at x = 1

d er i v a t i v e h (g (x) k (x))

d er i v a t i v e \frac{2}{3} (1 + x)^{\frac{3}{2}}

t an g e n t f (x) = 3 e^{x}, at x = 0