integral of 3tan^2(x)sec(x)

Lösung

\int 3 tan^{2} (x) sec (x) d x

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

Schritte zur Lösung

\int 3 tan^{2} (x) sec (x) d x

Entferne die Konstante:

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

= 3 \cdot \int tan^{2} (x) sec (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= 3 \cdot \int (- 1 + sec^{2} (x)) sec (x) d x

Multipliziere aus

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

= 3 \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

= 3 (- \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) ∣

= 3 (- 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) ∣

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

Füge eine Konstante zur Lösung hinzu

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

t an g e n t a x^{2} + b x + c

\frac{\partial}{\partial y} (sin (x) + π cos (y))

\int \frac{( 1 - y ) e ^{y}}{y ^{2}} d y

y (8 x - 9 y) d x + 2 x (x - 3 y) d y = 0

\frac{d}{d x} (a x \cdot e^{b x})