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

Lösung

\int_{0}^{2 π} cos^{4} (t) d t

\frac{3 π}{4}

Dezimale

2.35619 \dots

Schritte zur Lösung

\int_{0}^{2 π} cos^{4} (t) d t

Vereinfache

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

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

Wende die partielle Integration an

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

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

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

Vereinfache

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

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

Berechne die Grenzen:

\frac{3 π}{4}

= \frac{3 π}{4}

\int_{0}^{\frac{π}{4}} cos (2 t) d t

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