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Beliebt Rechnen >

integral of cos^4(ax)

Lösung

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

Lösung

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

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{cos ^{4} ( u )}{a} d u

Entferne die Konstante:

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

= \frac{1}{a} \cdot \int cos^{4} (u) d u

Vereinfache

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

= \frac{1}{a} \cdot \int cos^{3} (u) cos (u) d u

Wende die partielle Integration an

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

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

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

Setze in

u = a x

ein

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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