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

Lösung

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

\frac{1}{192} (12 x + 76 cos^{3} (2 x) sin (2 x) - 9 sin (8 x) - 16 cos^{5} (2 x) sin (2 x) - 15 sin (4 x)) + C

Schritte zur Lösung

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

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Multipliziere aus

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

= \frac{1}{2} \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}{2} (\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}{2} (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 = 2 x

ein

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

Vereinfache

\frac{1}{2} (cos^{3} (2 x) sin (2 x) + \frac{3}{8} (2 x - \frac{1}{4} sin (4 \cdot 2 x)) - (\frac{sin ( 2 x ) cos ^{5} ( 2 x )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (2 x) sin (2 x) + \frac{3}{8} (2 x + \frac{1}{2} sin (2 \cdot 2 x))))) : \frac{1}{192} (12 x + 76 cos^{3} (2 x) sin (2 x) - 9 sin (8 x) - 16 cos^{5} (2 x) sin (2 x) - 15 sin (4 x))

= \frac{1}{192} (12 x + 76 cos^{3} (2 x) sin (2 x) - 9 sin (8 x) - 16 cos^{5} (2 x) sin (2 x) - 15 sin (4 x))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{192} (12 x + 76 cos^{3} (2 x) sin (2 x) - 9 sin (8 x) - 16 cos^{5} (2 x) sin (2 x) - 15 sin (4 x)) + C

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

\frac{d}{d x} (- 2 a sin (2 x))

\frac{d}{d x} (x^{4} sin (x) tan (x))

t an g e n t - 4 x^{2} + 7 x

n = 3 \sum \infty e^{5 - 4 n}