integral of 8sin^4(x)cos^2(x)

Lösung

\int 8 sin^{4} (x) cos^{2} (x) d x

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

Schritte zur Lösung

\int 8 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

= 8 \cdot \int sin^{4} (x) cos^{2} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= 8 \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)

= 8 \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

= 8 (\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)))

= 8 (- \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

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

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

Füge eine Konstante zur Lösung hinzu

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

x \to 0 lim (\frac{2 tan ( x )}{x})

\frac{\partial}{\partial x} (6 x^{2} y)

d y = \frac{x + 6}{x + 1} d x

d er i v a t i v e y = 4 sin (5 x - 2)

\int cot^{6} (x) d x