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

Lösung

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

4 (\frac{3}{8} x - \frac{3}{32} sin (4 x) + \frac{19}{24} cos^{3} (x) sin (x) - \frac{1}{6} cos^{5} (x) sin (x) - \frac{5}{32} (2 x + sin (2 x))) + C

Schritte zur Lösung

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Entferne die Konstante:

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

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Multipliziere aus

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

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

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

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

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

Vereinfache

4 (cos^{3} (x) sin (x) + \frac{3}{8} (x - \frac{1}{4} sin (4 x)) - (\frac{sin ( x ) cos ^{5} ( x )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (x) sin (x) + \frac{3}{8} (x + \frac{1}{2} sin (2 x))))) : 4 (\frac{3}{8} x - \frac{3}{32} sin (4 x) + \frac{19}{24} cos^{3} (x) sin (x) - \frac{1}{6} cos^{5} (x) sin (x) - \frac{5}{32} (2 x + sin (2 x)))

= 4 (\frac{3}{8} x - \frac{3}{32} sin (4 x) + \frac{19}{24} cos^{3} (x) sin (x) - \frac{1}{6} cos^{5} (x) sin (x) - \frac{5}{32} (2 x + sin (2 x)))

Füge eine Konstante zur Lösung hinzu

= 4 (\frac{3}{8} x - \frac{3}{32} sin (4 x) + \frac{19}{24} cos^{3} (x) sin (x) - \frac{1}{6} cos^{5} (x) sin (x) - \frac{5}{32} (2 x + sin (2 x))) + C

x \to 0 lim ((1 - 10 x)^{\frac{1}{x}})

x \to \infty + lim (x^{(3)} e^{(- x^{(2)})})

\int x^{11} e^{x^{12}} d x

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

n = 1 \sum \infty 4^{- n}