integral from 0 to 2pi of cos^4(θ)

Lösung

\int_{0}^{2 π} cos^{4} (θ) d θ

\frac{3 π}{4}

Dezimale

2.35619 \dots

Schritte zur Lösung

\int_{0}^{2 π} cos^{4} (θ) d θ

Vereinfache

cos^{4} (θ) : cos^{3} (θ) cos (θ)

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

Wende die partielle Integration an

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

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

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

Vereinfache

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

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

Berechne die Grenzen:

\frac{3 π}{4}

= \frac{3 π}{4}

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