integral from 0 to 2pi of (cos(x))^4

Lösung

\int_{0}^{2 π} (cos (x))^{4} d x

\frac{3 π}{4}

Dezimale

2.35619 \dots

Schritte zur Lösung

\int_{0}^{2 π} (cos (x))^{4} d x

Vereinfache

= \int_{0}^{2 π} cos^{4} (x) d x

Vereinfache

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

= \int_{0}^{2 π} cos^{3} (x) cos (x) d x

Wende die partielle Integration an

= [cos^{3} (x) sin (x) - \int - 3 cos^{2} (x) sin^{2} (x) d x]_{0}^{2 π}

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

= [cos^{3} (x) sin (x) - (- \frac{3}{8} (x - \frac{1}{4} sin (4 x)))]_{0}^{2 π}

Vereinfache

[cos^{3} (x) sin (x) - (- \frac{3}{8} (x - \frac{1}{4} sin (4 x)))]_{0}^{2 π} : [\frac{1}{32} (32 cos^{3} (x) sin (x) + 3 (4 x - sin (4 x)))]_{0}^{2 π}

= [\frac{1}{32} (32 cos^{3} (x) sin (x) + 3 (4 x - sin (4 x)))]_{0}^{2 π}

Berechne die Grenzen:

\frac{3 π}{4}

= \frac{3 π}{4}

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