integral of (4-sec^2(3x))/(tan(3x))

Lösung

\int \frac{4 - sec ^{2} ( 3 x )}{tan ( 3 x )} d x

ln ∣ sin (3 x) ∣ + \frac{1}{3} ln ∣ cos (3 x) ∣ + C

Schritte zur Lösung

\int \frac{4 - sec ^{2} ( 3 x )}{tan ( 3 x )} d x

Multipliziere aus

\frac{4 - sec ^{2} ( 3 x )}{tan ( 3 x )} : \frac{4}{tan ( 3 x )} - \frac{sec ^{2} ( 3 x )}{tan ( 3 x )}

= \int \frac{4}{tan ( 3 x )} - \frac{sec ^{2} ( 3 x )}{tan ( 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 \frac{4}{tan ( 3 x )} d x - \int \frac{sec ^{2} ( 3 x )}{tan ( 3 x )} d x

\int \frac{4}{tan ( 3 x )} d x = \frac{4}{3} ln ∣ sin (3 x) ∣

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

= \frac{4}{3} ln ∣ sin (3 x) ∣ - (\frac{1}{3} ln ∣ sin (3 x) ∣ - \frac{1}{3} ln ∣ cos (3 x) ∣)

Vereinfache

\frac{4}{3} ln ∣ sin (3 x) ∣ - (\frac{1}{3} ln ∣ sin (3 x) ∣ - \frac{1}{3} ln ∣ cos (3 x) ∣) : ln ∣ sin (3 x) ∣ + \frac{1}{3} ln ∣ cos (3 x) ∣

= ln ∣ sin (3 x) ∣ + \frac{1}{3} ln ∣ cos (3 x) ∣

Füge eine Konstante zur Lösung hinzu

= ln ∣ sin (3 x) ∣ + \frac{1}{3} ln ∣ cos (3 x) ∣ + C

\int (x^{2} - 4)^{4} (2 x - 1) d x

\int x^{8} e^{x^{9} + 8} d x

\int \frac{6}{6 x - x ^{2} - 5} d x

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

\int \frac{cos ^{2} ( θ )}{sin ^{2} ( θ )} d θ