integral from 0 to pi/2 of cos^5(6x)

Lösung

\int_{0}^{\frac{π}{2}} cos^{5} (6 x) d x

0

Schritte zur Lösung

\int_{0}^{\frac{π}{2}} cos^{5} (6 x) d x

Wende U-Substitution an

= \int_{0}^{3 π} cos^{5} (u) \frac{1}{6} d u

Entferne die Konstante:

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

= \frac{1}{6} \cdot \int_{0}^{3 π} cos^{5} (u) d u

Wende integrale Reduktion an

= \frac{1}{6} ([\frac{sin ( u ) cos ^{4} ( u )}{5}]_{0}^{3 π} + \frac{4}{5} \cdot \int_{0}^{3 π} cos^{3} (u) d u)

\int_{0}^{3 π} cos^{3} (u) d u = 0

= \frac{1}{6} ([\frac{sin ( u ) cos ^{4} ( u )}{5}]_{0}^{3 π} + \frac{4}{5} \cdot 0)

Vereinfache

\frac{1}{6} ([\frac{sin ( u ) cos ^{4} ( u )}{5}]_{0}^{3 π} + \frac{4}{5} \cdot 0) : \frac{1}{6} [\frac{1}{5} cos^{4} (u) sin (u)]_{0}^{3 π}

= \frac{1}{6} [\frac{1}{5} cos^{4} (u) sin (u)]_{0}^{3 π}

Berechne die Grenzen:

0

= \frac{1}{6} \cdot 0

Vereinfache

= 0

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