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Beliebt Rechnen >

integral of (sin^4(cot(x)))/(sin^2(x))

Lösung

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

Lösung

\frac{1}{4} sin^{3} (cot (x)) cos (cot (x)) - \frac{3}{8} cot (x) + \frac{3}{16} sin (2 cot (x)) + C

Schritte zur Lösung

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

Wende U-Substitution an

= \int - sin^{4} (u) d u

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int sin^{4} (u) d u

Wende integrale Reduktion an

= - (- \frac{cos ( u ) sin ^{3} ( u )}{4} + \frac{3}{4} \cdot \int sin^{2} (u) d u)

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

= - (- \frac{cos ( u ) sin ^{3} ( u )}{4} + \frac{3}{4} \cdot \frac{1}{2} (u - \frac{1}{2} sin (2 u)))

Setze in

u = cot (x)

ein

= - (- \frac{cos ( cot ( x ) ) sin ^{3} ( cot ( x ) )}{4} + \frac{3}{4} \cdot \frac{1}{2} (cot (x) - \frac{1}{2} sin (2 cot (x))))

Vereinfache

- (- \frac{cos ( cot ( x ) ) sin ^{3} ( cot ( x ) )}{4} + \frac{3}{4} \cdot \frac{1}{2} (cot (x) - \frac{1}{2} sin (2 cot (x)))) : \frac{1}{4} sin^{3} (cot (x)) cos (cot (x)) - \frac{3}{8} cot (x) + \frac{3}{16} sin (2 cot (x))

= \frac{1}{4} sin^{3} (cot (x)) cos (cot (x)) - \frac{3}{8} cot (x) + \frac{3}{16} sin (2 cot (x))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{4} sin^{3} (cot (x)) cos (cot (x)) - \frac{3}{8} cot (x) + \frac{3}{16} sin (2 cot (x)) + C

Graph

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