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Beliebt Rechnen >

integral of (sin(3x)+3cos(3x))^2

Lösung

\int (sin (3 x) + 3 cos (3 x))^{2} d x

Lösung

5 x - \frac{1}{12} sin (6 x) - cos^{2} (3 x) + \frac{3}{4} sin (6 x) + C

Schritte zur Lösung

\int (sin (3 x) + 3 cos (3 x))^{2} d x

Multipliziere aus

(sin (3 x) + 3 cos (3 x))^{2} : sin^{2} (3 x) + 6 sin (3 x) cos (3 x) + 9 cos^{2} (3 x)

= \int sin^{2} (3 x) + 6 sin (3 x) cos (3 x) + 9 cos^{2} (3 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^{2} (3 x) d x + \int 6 sin (3 x) cos (3 x) d x + \int 9 cos^{2} (3 x) d x

\int sin^{2} (3 x) d x = \frac{1}{2} (x - \frac{1}{6} sin (6 x))

\int 6 sin (3 x) cos (3 x) d x = - cos^{2} (3 x)

\int 9 cos^{2} (3 x) d x = \frac{9}{2} (x + \frac{1}{6} sin (6 x))

= \frac{1}{2} (x - \frac{1}{6} sin (6 x)) - cos^{2} (3 x) + \frac{9}{2} (x + \frac{1}{6} sin (6 x))

Vereinfache

\frac{1}{2} (x - \frac{1}{6} sin (6 x)) - cos^{2} (3 x) + \frac{9}{2} (x + \frac{1}{6} sin (6 x)) : 5 x - \frac{1}{12} sin (6 x) - cos^{2} (3 x) + \frac{3}{4} sin (6 x)

= 5 x - \frac{1}{12} sin (6 x) - cos^{2} (3 x) + \frac{3}{4} sin (6 x)

Füge eine Konstante zur Lösung hinzu

= 5 x - \frac{1}{12} sin (6 x) - cos^{2} (3 x) + \frac{3}{4} sin (6 x) + C

Graph

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