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

Lösung

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

- \frac{1}{4} sin^{3} (x) cos (x) - \frac{3}{16} sin (2 x) + \frac{3}{32} sin (4 x) - cos^{3} (x) sin (x) + C

Schritte zur Lösung

\int sin^{4} (x) - cos^{4} (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

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

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

Vereinfache

- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x)) - (cos^{3} (x) sin (x) + \frac{3}{8} (x - \frac{1}{4} sin (4 x))) : - \frac{1}{4} sin^{3} (x) cos (x) - \frac{3}{16} sin (2 x) + \frac{3}{32} sin (4 x) - cos^{3} (x) sin (x)

= - \frac{1}{4} sin^{3} (x) cos (x) - \frac{3}{16} sin (2 x) + \frac{3}{32} sin (4 x) - cos^{3} (x) sin (x)

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{4} sin^{3} (x) cos (x) - \frac{3}{16} sin (2 x) + \frac{3}{32} sin (4 x) - cos^{3} (x) sin (x) + C

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