integral of (1+cos(2x))^2

Lösung

\int (1 + cos (2 x))^{2} d x

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

Schritte zur Lösung

\int (1 + cos (2 x))^{2} d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 4 \cdot \int cos^{4} (x) d x

Vereinfache

cos^{4} (x) : cos^{3} (x) cos (x)

= 4 \cdot \int cos^{3} (x) cos (x) d x

Wende die partielle Integration an

= 4 (cos^{3} (x) sin (x) - \int - 3 cos^{2} (x) sin^{2} (x) d x)

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

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