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Beliebt Rechnen >

integral of (cos(x))/(2sin^2(x)+3sin(x))

Lösung

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

Lösung

\frac{1}{3} ln ∣ sin (x) ∣ - \frac{1}{3} ln ∣ 2 sin (x) + 3 ∣ + C

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{1}{2 u ^{2} + 3 u} d u

Ermittle den Partialbruch von

\frac{1}{2 u ^{2} + 3 u} : \frac{1}{3 u} - \frac{2}{3 ( 2 u + 3 )}

= \int \frac{1}{3 u} - \frac{2}{3 ( 2 u + 3 )} 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}{3 u} d u - \int \frac{2}{3 ( 2 u + 3 )} d u

\int \frac{1}{3 u} d u = \frac{1}{3} ln ∣ u ∣

\int \frac{2}{3 ( 2 u + 3 )} d u = \frac{1}{3} ln ∣ 2 u + 3 ∣

= \frac{1}{3} ln ∣ u ∣ - \frac{1}{3} ln ∣ 2 u + 3 ∣

Setze in

u = sin (x)

ein

= \frac{1}{3} ln ∣ sin (x) ∣ - \frac{1}{3} ln ∣ 2 sin (x) + 3 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{1}{3} ln ∣ sin (x) ∣ - \frac{1}{3} ln ∣ 2 sin (x) + 3 ∣ + C

Graph

Beliebte Beispiele

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

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

integral of e^xsqrt(16-e^{2x)}

\int e^{x} 16 - e^{2 x} d x

integral of x^2e^{x^3-3}

\int x^{2} e^{x^{3} - 3} d x

integral of x^3+xy^2

\int x^{3} + x y^{2} d x

integral of (4-7x)\sqrt[3]{x}

\int (4 - 7 x) 3 x d x