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

Lösung

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

\frac{x ^{3} cos ^{2} ( a )}{3} + \frac{1}{3} x^{3} sin (2 a) + \frac{x ^{3} sin ^{2} ( a )}{3} + C

Schritte zur Lösung

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

Multipliziere aus

(cos (a) x + sin (a) x)^{2} : x^{2} cos^{2} (a) + 2 x^{2} cos (a) sin (a) + x^{2} sin^{2} (a)

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

\int x^{2} cos^{2} (a) d x = \frac{x ^{3} cos ^{2} ( a )}{3}

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

\int x^{2} sin^{2} (a) d x = \frac{x ^{3} sin ^{2} ( a )}{3}

= \frac{x ^{3} cos ^{2} ( a )}{3} + \frac{1}{3} x^{3} sin (2 a) + \frac{x ^{3} sin ^{2} ( a )}{3}

Füge eine Konstante zur Lösung hinzu

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

\int \frac{sin ( 3 x )}{2} d x

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