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Beliebt Rechnen >

integral of (cos(x))/(sin^2(x)-2sin(x)-8)

Lösung

\int \frac{cos ( x )}{sin ^{2} ( x ) - 2 sin ( x ) - 8} d x

Lösung

- \frac{1}{6} ln ∣ sin (x) + 2 ∣ + \frac{1}{6} ln ∣ sin (x) - 4 ∣ + C

Schritte zur Lösung

\int \frac{cos ( x )}{sin ^{2} ( x ) - 2 sin ( x ) - 8} d x

Faktorisiere

\frac{cos ( x )}{sin ^{2} ( x ) - 2 sin ( x ) - 8}

= \int \frac{cos ( x )}{( sin ( x ) + 2 ) ( sin ( x ) - 4 )} d x

Wende U-Substitution an

= \int \frac{1}{( u + 2 ) ( u - 4 )} d u

Ermittle den Partialbruch von

\frac{1}{( u + 2 ) ( u - 4 )} : - \frac{1}{6 ( u + 2 )} + \frac{1}{6 ( u - 4 )}

= \int - \frac{1}{6 ( u + 2 )} + \frac{1}{6 ( u - 4 )} 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}{6 ( u + 2 )} d u + \int \frac{1}{6 ( u - 4 )} d u

\int \frac{1}{6 ( u + 2 )} d u = \frac{1}{6} ln ∣ u + 2 ∣

\int \frac{1}{6 ( u - 4 )} d u = \frac{1}{6} ln ∣ u - 4 ∣

= - \frac{1}{6} ln ∣ u + 2 ∣ + \frac{1}{6} ln ∣ u - 4 ∣

Setze in

u = sin (x)

ein

= - \frac{1}{6} ln ∣ sin (x) + 2 ∣ + \frac{1}{6} ln ∣ sin (x) - 4 ∣

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{6} ln ∣ sin (x) + 2 ∣ + \frac{1}{6} ln ∣ sin (x) - 4 ∣ + C

Graph

Beliebte Beispiele

integral of sec((4x)/3)

\int sec (\frac{4 x}{3}) d x

integral of (sin(2x))/(1+sin^4(x))

\int \frac{sin ( 2 x )}{1 + sin ^{4} ( x )} d x

integral of (x^3)/((1+x^2)^2)

\int \frac{x ^{3}}{( 1 + x ^{2} ) ^{2}} d x

integral of (x^5)/(1+x^2)

\int \frac{x ^{5}}{1 + x ^{2}} d x

integral of 3x+sin(x)

\int 3 x + sin (x) d x