integral of 10tan^2(x)sec(x)

Lösung

\int 10 tan^{2} (x) sec (x) d x

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

Schritte zur Lösung

\int 10 tan^{2} (x) sec (x) d x

Entferne die Konstante:

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

= 10 \cdot \int tan^{2} (x) sec (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= 10 \cdot \int (- 1 + sec^{2} (x)) sec (x) d x

Multipliziere aus

(- 1 + sec^{2} (x)) sec (x) : - sec (x) + sec^{3} (x)

= 10 \cdot \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

= 10 (- \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) ∣

= 10 (- 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{1}{2} ln ∣ tan (x) + sec (x) ∣

= 10 (\frac{1}{2} sec (x) tan (x) - \frac{1}{2} ln ∣ tan (x) + sec (x) ∣)

Füge eine Konstante zur Lösung hinzu

= 10 (\frac{1}{2} sec (x) tan (x) - \frac{1}{2} ln ∣ tan (x) + sec (x) ∣) + C

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