integral of [sec(x)+sec^3(x)]

Lösung

\int [sec (x) + sec^{3} (x)] d x

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

Schritte zur Lösung

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

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int sec (x) d x + \int sec^{3} (x) d x

\int sec (x) d x = ln ∣ tan (x) + sec (x) ∣

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

= ln ∣ tan (x) + sec (x) ∣ + \frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣

Vereinfache

ln ∣ tan (x) + sec (x) ∣ + \frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣ : \frac{1}{2} sec (x) tan (x) + \frac{3}{2} ln ∣ tan (x) + sec (x) ∣

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

Füge eine Konstante zur Lösung hinzu

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

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