integral of cos^4(4x)*sin^2(4x)

Lösung

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

\frac{1}{384} (24 x + 76 cos^{3} (4 x) sin (4 x) - 9 sin (16 x) - 16 cos^{5} (4 x) sin (4 x) - 15 sin (8 x)) + C

Schritte zur Lösung

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

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Multipliziere aus

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

= \frac{1}{4} \cdot \int cos^{4} (u) - cos^{6} (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

= \frac{1}{4} (\int cos^{4} (u) d u - \int cos^{6} (u) d u)

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

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

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

Setze in

u = 4 x

ein

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

Vereinfache

\frac{1}{4} (cos^{3} (4 x) sin (4 x) + \frac{3}{8} (4 x - \frac{1}{4} sin (4 \cdot 4 x)) - (\frac{sin ( 4 x ) cos ^{5} ( 4 x )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (4 x) sin (4 x) + \frac{3}{8} (4 x + \frac{1}{2} sin (2 \cdot 4 x))))) : \frac{1}{384} (24 x + 76 cos^{3} (4 x) sin (4 x) - 9 sin (16 x) - 16 cos^{5} (4 x) sin (4 x) - 15 sin (8 x))

= \frac{1}{384} (24 x + 76 cos^{3} (4 x) sin (4 x) - 9 sin (16 x) - 16 cos^{5} (4 x) sin (4 x) - 15 sin (8 x))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{384} (24 x + 76 cos^{3} (4 x) sin (4 x) - 9 sin (16 x) - 16 cos^{5} (4 x) sin (4 x) - 15 sin (8 x)) + C

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