integral of (cos(x))/(sin^2(x)-sin(x))

Lösung

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

- ln ∣ sin (x) ∣ + ln ∣ sin (x) - 1 ∣ + C

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{1}{u ^{2} - u} d u

Ermittle den Partialbruch von

\frac{1}{u ^{2} - u} : - \frac{1}{u} + \frac{1}{u - 1}

= \int - \frac{1}{u} + \frac{1}{u - 1} 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

= - \int \frac{1}{u} d u + \int \frac{1}{u - 1} d u

\int \frac{1}{u} d u = ln ∣ u ∣

\int \frac{1}{u - 1} d u = ln ∣ u - 1 ∣

= - ln ∣ u ∣ + ln ∣ u - 1 ∣

Setze in

u = sin (x)

ein

= - ln ∣ sin (x) ∣ + ln ∣ sin (x) - 1 ∣

Füge eine Konstante zur Lösung hinzu

= - ln ∣ sin (x) ∣ + ln ∣ sin (x) - 1 ∣ + C

\int 3 x^{2} \cdot cos (x) d x

\int \frac{1}{3} sin^{3} (x) cos^{3} (x) d x

\int 2 ln (3 x + 1) d x

\int \frac{e ^{3 l n (x)}}{7 x ^{3}} d x

\int \frac{1}{x ^{2} 9 + 4 x ^{2}} d x