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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Wende U-Substitution an

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

Ermittle den Partialbruch von

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

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

\int \frac{1}{u} d u = ln ∣ u ∣

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

= ln ∣ u ∣ - \frac{1}{2} ln u^{2} + 1

Setze in

u = sin (x)

ein

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of tsin(4)t^2

\int t sin (4) t^{2} d t

integral of ((e^x-1))/(e^x+1)

\int \frac{( e ^{x} - 1 )}{e ^{x} + 1} d x

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

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integral of (arccot(sqrt(x)))/(sqrt(x))

\int \frac{arccot ( x )}{x} d x