integral of sin^2(5x)cos^4(5x)

Lösung

\int sin^{2} (5 x) cos^{4} (5 x) d x

\frac{1}{480} (30 x + 76 cos^{3} (5 x) sin (5 x) - 9 sin (20 x) - 16 cos^{5} (5 x) sin (5 x) - 15 sin (10 x)) + C

Schritte zur Lösung

\int sin^{2} (5 x) cos^{4} (5 x) d x

Wende U-Substitution an

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

Entferne die Konstante:

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

= \frac{1}{5} \cdot \int sin^{2} (u) cos^{4} (u) d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \frac{1}{5} \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}{5} \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}{5} (\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}{5} (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 = 5 x

ein

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

Vereinfache

\frac{1}{5} (cos^{3} (5 x) sin (5 x) + \frac{3}{8} (5 x - \frac{1}{4} sin (4 \cdot 5 x)) - (\frac{sin ( 5 x ) cos ^{5} ( 5 x )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (5 x) sin (5 x) + \frac{3}{8} (5 x + \frac{1}{2} sin (2 \cdot 5 x))))) : \frac{1}{480} (30 x + 76 cos^{3} (5 x) sin (5 x) - 9 sin (20 x) - 16 cos^{5} (5 x) sin (5 x) - 15 sin (10 x))

= \frac{1}{480} (30 x + 76 cos^{3} (5 x) sin (5 x) - 9 sin (20 x) - 16 cos^{5} (5 x) sin (5 x) - 15 sin (10 x))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{480} (30 x + 76 cos^{3} (5 x) sin (5 x) - 9 sin (20 x) - 16 cos^{5} (5 x) sin (5 x) - 15 sin (10 x)) + C