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

Lösung

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

2 (- \frac{1}{4} sin^{3} (\frac{x}{2}) cos (\frac{x}{2}) + \frac{3}{16} (x - sin (x)) - \frac{3}{256} (5 (3 (x - sin (x)) - 4 sin^{3} (\frac{x}{2}) cos (\frac{x}{2})) - 16 sin^{5} (\frac{x}{2}) cos (\frac{x}{2})) - \frac{1}{8} sin^{7} (\frac{x}{2}) cos (\frac{x}{2})) + C

Schritte zur Lösung

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

Vereinfache

sin^{4} (\frac{1}{2} x) cos^{4} (\frac{1}{2} x) : sin^{4} (\frac{x}{2}) cos^{4} (\frac{x}{2})

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

Wende U-Substitution an

= \int sin^{4} (u) cos^{4} (u) \cdot 2 d u

Entferne die Konstante:

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

= 2 \cdot \int sin^{4} (u) cos^{4} (u) d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= 2 \cdot \int sin^{4} (u) (1 - sin^{2} (u))^{2} d u

Multipliziere aus

sin^{4} (u) (1 - sin^{2} (u))^{2} : sin^{4} (u) - 2 sin^{6} (u) + sin^{8} (u)

= 2 \cdot \int sin^{4} (u) - 2 sin^{6} (u) + sin^{8} (u) 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

= 2 (\int sin^{4} (u) d u - \int 2 sin^{6} (u) d u + \int sin^{8} (u) d u)

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

\int 2 sin^{6} (u) d u = 2 (- \frac{cos ( u ) sin ^{5} ( u )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (u) cos (u) + \frac{3}{8} (u - \frac{1}{2} sin (2 u))))

\int sin^{8} (u) d u = - \frac{cos ( u ) sin ^{7} ( u )}{8} + \frac{7}{8} (- \frac{cos ( u ) sin ^{5} ( u )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (u) cos (u) + \frac{3}{8} (u - \frac{1}{2} sin (2 u))))

= 2 (- \frac{1}{4} sin^{3} (u) cos (u) + \frac{3}{8} (u - \frac{1}{2} sin (2 u)) - 2 (- \frac{cos ( u ) sin ^{5} ( u )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (u) cos (u) + \frac{3}{8} (u - \frac{1}{2} sin (2 u)))) - \frac{cos ( u ) sin ^{7} ( u )}{8} + \frac{7}{8} (- \frac{cos ( u ) sin ^{5} ( u )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (u) cos (u) + \frac{3}{8} (u - \frac{1}{2} sin (2 u)))))

Setze in

u = \frac{x}{2}

ein

= 2 (- \frac{1}{4} sin^{3} (\frac{x}{2}) cos (\frac{x}{2}) + \frac{3}{8} (\frac{x}{2} - \frac{1}{2} sin (2 \cdot \frac{x}{2})) - 2 (- \frac{cos ( \frac{x}{2} ) sin ^{5} ( \frac{x}{2} )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (\frac{x}{2}) cos (\frac{x}{2}) + \frac{3}{8} (\frac{x}{2} - \frac{1}{2} sin (2 \cdot \frac{x}{2})))) - \frac{cos ( \frac{x}{2} ) sin ^{7} ( \frac{x}{2} )}{8} + \frac{7}{8} (- \frac{cos ( \frac{x}{2} ) sin ^{5} ( \frac{x}{2} )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (\frac{x}{2}) cos (\frac{x}{2}) + \frac{3}{8} (\frac{x}{2} - \frac{1}{2} sin (2 \cdot \frac{x}{2})))))

Vereinfache

2 (- \frac{1}{4} sin^{3} (\frac{x}{2}) cos (\frac{x}{2}) + \frac{3}{8} (\frac{x}{2} - \frac{1}{2} sin (2 \cdot \frac{x}{2})) - 2 (- \frac{cos ( \frac{x}{2} ) sin ^{5} ( \frac{x}{2} )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (\frac{x}{2}) cos (\frac{x}{2}) + \frac{3}{8} (\frac{x}{2} - \frac{1}{2} sin (2 \cdot \frac{x}{2})))) - \frac{cos ( \frac{x}{2} ) sin ^{7} ( \frac{x}{2} )}{8} + \frac{7}{8} (- \frac{cos ( \frac{x}{2} ) sin ^{5} ( \frac{x}{2} )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (\frac{x}{2}) cos (\frac{x}{2}) + \frac{3}{8} (\frac{x}{2} - \frac{1}{2} sin (2 \cdot \frac{x}{2}))))) : 2 (- \frac{1}{4} sin^{3} (\frac{x}{2}) cos (\frac{x}{2}) + \frac{3}{16} (x - sin (x)) - \frac{3}{256} (5 (3 (x - sin (x)) - 4 sin^{3} (\frac{x}{2}) cos (\frac{x}{2})) - 16 sin^{5} (\frac{x}{2}) cos (\frac{x}{2})) - \frac{1}{8} sin^{7} (\frac{x}{2}) cos (\frac{x}{2}))

= 2 (- \frac{1}{4} sin^{3} (\frac{x}{2}) cos (\frac{x}{2}) + \frac{3}{16} (x - sin (x)) - \frac{3}{256} (5 (3 (x - sin (x)) - 4 sin^{3} (\frac{x}{2}) cos (\frac{x}{2})) - 16 sin^{5} (\frac{x}{2}) cos (\frac{x}{2})) - \frac{1}{8} sin^{7} (\frac{x}{2}) cos (\frac{x}{2}))

Füge eine Konstante zur Lösung hinzu

= 2 (- \frac{1}{4} sin^{3} (\frac{x}{2}) cos (\frac{x}{2}) + \frac{3}{16} (x - sin (x)) - \frac{3}{256} (5 (3 (x - sin (x)) - 4 sin^{3} (\frac{x}{2}) cos (\frac{x}{2})) - 16 sin^{5} (\frac{x}{2}) cos (\frac{x}{2})) - \frac{1}{8} sin^{7} (\frac{x}{2}) cos (\frac{x}{2})) + C