integral of (sec(3x)+csc(3x))^2

Lösung

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

\frac{1}{3} (tan (3 x) + ln (tan^{2} (3 x)) - cot (3 x)) + C

Schritte zur Lösung

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

Multipliziere aus

(sec (3 x) + csc (3 x))^{2} : sec^{2} (3 x) + 2 sec (3 x) csc (3 x) + csc^{2} (3 x)

= \int sec^{2} (3 x) + 2 sec (3 x) csc (3 x) + csc^{2} (3 x) d x

Wende U-Substitution an

= \int (sec^{2} (u) + 2 sec (u) csc (u) + csc^{2} (u)) \frac{1}{3} d u

Entferne die Konstante:

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

= \frac{1}{3} \cdot \int sec^{2} (u) + 2 sec (u) csc (u) + csc^{2} (u) d u

Wende die Summenregel an:

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

= \frac{1}{3} (\int sec^{2} (u) d u + \int 2 sec (u) csc (u) d u + \int csc^{2} (u) d u)

\int sec^{2} (u) d u = tan (u)

\int 2 sec (u) csc (u) d u = ln (tan^{2} (u))

\int csc^{2} (u) d u = - cot (u)

= \frac{1}{3} (tan (u) + ln (tan^{2} (u)) - cot (u))

Setze in

u = 3 x

ein

= \frac{1}{3} (tan (3 x) + ln (tan^{2} (3 x)) - cot (3 x))

Füge eine Konstante zur Lösung hinzu

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

\int \frac{x ^{2} - 4}{x ^{3} - 3 x ^{2} - x + 3} d x

\int \frac{x}{3 x ^{2} + 9} d x

d er i v a t i v e (4 - t) (4 + t^{3})^{- 1}

\frac{\partial}{\partial y} (\frac{4}{3} π (x^{3} - y^{3}))

x \to \infty lim (\frac{3}{5 x})