integral of 3sec(x)(tan(x)-sec(x))

Lösung

\int 3 sec (x) (tan (x) - sec (x)) d x

3 (sec (x) - tan (x)) + C

Schritte zur Lösung

\int 3 sec (x) (tan (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 sec (x) (tan (x) - sec (x)) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= 3 \cdot \int \frac{sin ( x ) - 1}{cos ^{2} ( x )} d x

Multipliziere aus

\frac{sin ( x ) - 1}{cos ^{2} ( x )} : \frac{sin ( x )}{cos ^{2} ( x )} - \frac{1}{cos ^{2} ( x )}

= 3 \cdot \int \frac{sin ( x )}{cos ^{2} ( x )} - \frac{1}{cos ^{2} ( 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 \frac{sin ( x )}{cos ^{2} ( x )} d x - \int \frac{1}{cos ^{2} ( x )} d x)

\int \frac{sin ( x )}{cos ^{2} ( x )} d x = sec (x)

\int \frac{1}{cos ^{2} ( x )} d x = tan (x)

= 3 (sec (x) - tan (x))

Füge eine Konstante zur Lösung hinzu

= 3 (sec (x) - tan (x)) + C

d er i v a t i v e g (t) = 3 t + 5

d er i v a t i v e f (x) = \frac{6}{5 x + 2}

\int \frac{1}{9 x + 5} d x

\frac{\partial}{\partial x} (a x)

d er i v a t i v e 4 ∣ x + 3 ∣ - 5