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

Lösung

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

5 x + 2 sin (2 x) + 3 sin^{2} (x) + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

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

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

Vereinfache

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

= 5 x + 2 sin (2 x) + 3 sin^{2} (x)

Füge eine Konstante zur Lösung hinzu

= 5 x + 2 sin (2 x) + 3 sin^{2} (x) + C