integral from-pi/2 to pi/2 of 4cos^6(θ)

Lösung

\int_{- \frac{π}{2}}^{\frac{π}{2}} 4 cos^{6} (θ) d θ

\frac{5 π}{4}

Dezimale

3.92699 \dots

Schritte zur Lösung

\int_{- \frac{π}{2}}^{\frac{π}{2}} 4 cos^{6} (θ) d θ

Entferne die Konstante:

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

= 4 \cdot \int_{- \frac{π}{2}}^{\frac{π}{2}} cos^{6} (θ) d θ

Wende integrale Reduktion an

= 4 ([\frac{sin ( θ ) cos ^{5} ( θ )}{6}]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{5}{6} \cdot \int_{- \frac{π}{2}}^{\frac{π}{2}} cos^{4} (θ) d θ)

\int_{- \frac{π}{2}}^{\frac{π}{2}} cos^{4} (θ) d θ = \frac{3 π}{8}

= 4 ([\frac{sin ( θ ) cos ^{5} ( θ )}{6}]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{5}{6} \cdot \frac{3 π}{8})

Vereinfache

4 ([\frac{sin ( θ ) cos ^{5} ( θ )}{6}]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{5}{6} \cdot \frac{3 π}{8}) : 4 ([\frac{1}{6} cos^{5} (θ) sin (θ)]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{5 π}{16})

= 4 ([\frac{1}{6} cos^{5} (θ) sin (θ)]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{5 π}{16})

Berechne die Grenzen:

0

= 4 (0 + \frac{5 π}{16})

Vereinfache

= \frac{5 π}{4}

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