integral of sin^6((3θ)/2)

Lösung

\int sin^{6} (\frac{3 θ}{2}) d θ

\frac{1}{144} (- 16 sin^{5} (\frac{3 θ}{2}) cos (\frac{3 θ}{2}) + 5 (3 (3 θ - sin (3 θ)) - 4 sin^{3} (\frac{3 θ}{2}) cos (\frac{3 θ}{2}))) + C

Schritte zur Lösung

\int sin^{6} (\frac{3 θ}{2}) d θ

Wende U-Substitution an

= \int sin^{6} (\frac{u}{2}) \frac{1}{3} d u

Entferne die Konstante:

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

= \frac{1}{3} \cdot \int sin^{6} (\frac{u}{2}) d u

Wende U-Substitution an

= \frac{1}{3} \cdot \int sin^{6} (v) \cdot 2 d v

Entferne die Konstante:

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

= \frac{1}{3} \cdot 2 \cdot \int sin^{6} (v) d v

Wende integrale Reduktion an

= \frac{1}{3} \cdot 2 (- \frac{cos ( v ) sin ^{5} ( v )}{6} + \frac{5}{6} \cdot \int sin^{4} (v) d v)

\int sin^{4} (v) d v = - \frac{1}{4} sin^{3} (v) cos (v) + \frac{3}{8} (v - \frac{1}{2} sin (2 v))

= \frac{1}{3} \cdot 2 (- \frac{cos ( v ) sin ^{5} ( v )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (v) cos (v) + \frac{3}{8} (v - \frac{1}{2} sin (2 v))))

Ersetze zurück

= \frac{1}{3} \cdot 2 (- \frac{cos ( \frac{3 θ}{2} ) sin ^{5} ( \frac{3 θ}{2} )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (\frac{3 θ}{2}) cos (\frac{3 θ}{2}) + \frac{3}{8} (\frac{3 θ}{2} - \frac{1}{2} sin (2 \cdot \frac{3 θ}{2}))))

Vereinfache

\frac{1}{3} \cdot 2 (- \frac{cos ( \frac{3 θ}{2} ) sin ^{5} ( \frac{3 θ}{2} )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (\frac{3 θ}{2}) cos (\frac{3 θ}{2}) + \frac{3}{8} (\frac{3 θ}{2} - \frac{1}{2} sin (2 \cdot \frac{3 θ}{2})))) : \frac{1}{144} (- 16 sin^{5} (\frac{3 θ}{2}) cos (\frac{3 θ}{2}) + 5 (3 (3 θ - sin (3 θ)) - 4 sin^{3} (\frac{3 θ}{2}) cos (\frac{3 θ}{2})))

= \frac{1}{144} (- 16 sin^{5} (\frac{3 θ}{2}) cos (\frac{3 θ}{2}) + 5 (3 (3 θ - sin (3 θ)) - 4 sin^{3} (\frac{3 θ}{2}) cos (\frac{3 θ}{2})))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{144} (- 16 sin^{5} (\frac{3 θ}{2}) cos (\frac{3 θ}{2}) + 5 (3 (3 θ - sin (3 θ)) - 4 sin^{3} (\frac{3 θ}{2}) cos (\frac{3 θ}{2}))) + C

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