integral of sin^2(x)sin^2(2x)

Lösung

\int sin^{2} (x) sin^{2} (2 x) d x

4 (\frac{3}{8} x - \frac{1}{24} sin^{3} (x) cos (x) - \frac{3}{16} sin (2 x) + \frac{1}{6} sin^{5} (x) cos (x) - \frac{5}{32} (2 x - sin (2 x))) + C

Schritte zur Lösung

\int sin^{2} (x) sin^{2} (2 x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Entferne die Konstante:

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

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Multipliziere aus

sin^{4} (x) (1 - sin^{2} (x)) : sin^{4} (x) - sin^{6} (x)

= 4 \cdot \int sin^{4} (x) - sin^{6} (x) d x

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 4 (\int sin^{4} (x) d x - \int sin^{6} (x) d x)

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

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

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

Vereinfache

4 (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x)) - (- \frac{cos ( x ) sin ^{5} ( x )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x))))) : 4 (\frac{3}{8} x - \frac{1}{24} sin^{3} (x) cos (x) - \frac{3}{16} sin (2 x) + \frac{1}{6} sin^{5} (x) cos (x) - \frac{5}{32} (2 x - sin (2 x)))

= 4 (\frac{3}{8} x - \frac{1}{24} sin^{3} (x) cos (x) - \frac{3}{16} sin (2 x) + \frac{1}{6} sin^{5} (x) cos (x) - \frac{5}{32} (2 x - sin (2 x)))

Füge eine Konstante zur Lösung hinzu

= 4 (\frac{3}{8} x - \frac{1}{24} sin^{3} (x) cos (x) - \frac{3}{16} sin (2 x) + \frac{1}{6} sin^{5} (x) cos (x) - \frac{5}{32} (2 x - sin (2 x))) + C

\int 2 e d x

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