integral of sec^7(5x+4)

Lösung

\int sec^{7} (5 x + 4) d x

\frac{1}{240} (8 sec^{6} (4 + 5 x) sin (4 + 5 x) + 5 (2 sec^{4} (4 + 5 x) sin (4 + 5 x) + 3 (sec (4 + 5 x) tan (4 + 5 x) + ln ∣ tan (4 + 5 x) + sec (4 + 5 x) ∣))) + C

Schritte zur Lösung

\int sec^{7} (5 x + 4) d x

Wende U-Substitution an

= \int sec^{7} (u) \frac{1}{5} d u

Entferne die Konstante:

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

= \frac{1}{5} \cdot \int sec^{7} (u) d u

Wende integrale Reduktion an

= \frac{1}{5} (\frac{sec ^{6} ( u ) sin ( u )}{6} + \frac{5}{6} \cdot \int sec^{5} (u) d u)

\int sec^{5} (u) d u = \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) ∣)

= \frac{1}{5} (\frac{sec ^{6} ( u ) sin ( u )}{6} + \frac{5}{6} (\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 = 4 + 5 x

ein

= \frac{1}{5} (\frac{sec ^{6} ( 4 + 5 x ) sin ( 4 + 5 x )}{6} + \frac{5}{6} (\frac{sec ^{4} ( 4 + 5 x ) sin ( 4 + 5 x )}{4} + \frac{3}{4} (\frac{1}{2} sec (4 + 5 x) tan (4 + 5 x) + \frac{1}{2} ln ∣ tan (4 + 5 x) + sec (4 + 5 x) ∣)))

Vereinfache

\frac{1}{5} (\frac{sec ^{6} ( 4 + 5 x ) sin ( 4 + 5 x )}{6} + \frac{5}{6} (\frac{sec ^{4} ( 4 + 5 x ) sin ( 4 + 5 x )}{4} + \frac{3}{4} (\frac{1}{2} sec (4 + 5 x) tan (4 + 5 x) + \frac{1}{2} ln ∣ tan (4 + 5 x) + sec (4 + 5 x) ∣))) : \frac{1}{240} (8 sec^{6} (4 + 5 x) sin (4 + 5 x) + 5 (2 sec^{4} (4 + 5 x) sin (4 + 5 x) + 3 (sec (4 + 5 x) tan (4 + 5 x) + ln ∣ tan (4 + 5 x) + sec (4 + 5 x) ∣)))

= \frac{1}{240} (8 sec^{6} (4 + 5 x) sin (4 + 5 x) + 5 (2 sec^{4} (4 + 5 x) sin (4 + 5 x) + 3 (sec (4 + 5 x) tan (4 + 5 x) + ln ∣ tan (4 + 5 x) + sec (4 + 5 x) ∣)))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{240} (8 sec^{6} (4 + 5 x) sin (4 + 5 x) + 5 (2 sec^{4} (4 + 5 x) sin (4 + 5 x) + 3 (sec (4 + 5 x) tan (4 + 5 x) + ln ∣ tan (4 + 5 x) + sec (4 + 5 x) ∣))) + C

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

x \to 4 + lim (\frac{7}{( x - 4 ) ^{2}})

\frac{d}{d y} ((cos (x) + sin (y))^{y s i n (2 x - y)})

t a y l or 2 x ln (x + 1)

d er i v a t i v e y = \frac{200}{0.9 + 0.1 e ^{2 x}}