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

Lösung

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

\frac{13}{2} x - \frac{5}{4} sin (2 x) + 6 sin^{2} (x) + C

Schritte zur Lösung

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

Multipliziere aus

(3 sin (x) + 2 cos (x))^{2} : 9 sin^{2} (x) + 12 sin (x) cos (x) + 4 cos^{2} (x)

= \int 9 sin^{2} (x) + 12 sin (x) cos (x) + 4 cos^{2} (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 9 sin^{2} (x) d x + \int 12 sin (x) cos (x) d x + \int 4 cos^{2} (x) d x

\int 9 sin^{2} (x) d x = \frac{9}{2} (x - \frac{1}{2} sin (2 x))

\int 12 sin (x) cos (x) d x = 6 sin^{2} (x)

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

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

Vereinfache

\frac{9}{2} (x - \frac{1}{2} sin (2 x)) + 6 sin^{2} (x) + 2 x + sin (2 x) : \frac{13}{2} x - \frac{5}{4} sin (2 x) + 6 sin^{2} (x)

= \frac{13}{2} x - \frac{5}{4} sin (2 x) + 6 sin^{2} (x)

Füge eine Konstante zur Lösung hinzu

= \frac{13}{2} x - \frac{5}{4} sin (2 x) + 6 sin^{2} (x) + C

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