integral of sec^7(x)

Lösung

\int sec^{7} (x) d x

\frac{1}{6} sec^{6} (x) sin (x) + \frac{5}{48} (2 sec^{4} (x) sin (x) + 3 (sec (x) tan (x) + ln ∣ tan (x) + sec (x) ∣)) + C

Schritte zur Lösung

\int sec^{7} (x) d x

Wende integrale Reduktion an

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

\int sec^{5} (x) d x = \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) ∣)

= \frac{sec ^{6} ( x ) sin ( x )}{6} + \frac{5}{6} (\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) ∣))

Vereinfache

\frac{sec ^{6} ( x ) sin ( x )}{6} + \frac{5}{6} (\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) ∣)) : \frac{1}{6} sec^{6} (x) sin (x) + \frac{5}{48} (2 sec^{4} (x) sin (x) + 3 (sec (x) tan (x) + ln ∣ tan (x) + sec (x) ∣))

= \frac{1}{6} sec^{6} (x) sin (x) + \frac{5}{48} (2 sec^{4} (x) sin (x) + 3 (sec (x) tan (x) + ln ∣ tan (x) + sec (x) ∣))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{6} sec^{6} (x) sin (x) + \frac{5}{48} (2 sec^{4} (x) sin (x) + 3 (sec (x) tan (x) + ln ∣ tan (x) + sec (x) ∣)) + C

\frac{d y}{d x} = cot (x + y + 3) - 1

\int \frac{e ^{2 x} - 1}{e ^{2 x} + 1} d x

x \to 6 + lim (\frac{8 + x}{6 - x})

\int x^{3} e^{(x^{2})} d x

\int (8 x + 3) d x