integral from 0 to pi/6 of (cos(3x))^8

Lösung

\int_{0}^{\frac{π}{6}} (cos (3 x))^{8} d x

\frac{35 π}{768}

Dezimale

0.14317 \dots

Schritte zur Lösung

\int_{0}^{\frac{π}{6}} (cos (3 x))^{8} d x

Vereinfache

= \int_{0}^{\frac{π}{6}} cos^{8} (3 x) d x

Wende U-Substitution an

= \int_{0}^{\frac{π}{2}} cos^{8} (u) \frac{1}{3} d u

Entferne die Konstante:

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

= \frac{1}{3} \cdot \int_{0}^{\frac{π}{2}} cos^{8} (u) d u

Wende integrale Reduktion an

= \frac{1}{3} ([\frac{sin ( u ) cos ^{7} ( u )}{8}]_{0}^{\frac{π}{2}} + \frac{7}{8} \cdot \int_{0}^{\frac{π}{2}} cos^{6} (u) d u)

\int_{0}^{\frac{π}{2}} cos^{6} (u) d u = \frac{5 π}{32}

= \frac{1}{3} ([\frac{sin ( u ) cos ^{7} ( u )}{8}]_{0}^{\frac{π}{2}} + \frac{7}{8} \cdot \frac{5 π}{32})

Vereinfache

\frac{1}{3} ([\frac{sin ( u ) cos ^{7} ( u )}{8}]_{0}^{\frac{π}{2}} + \frac{7}{8} \cdot \frac{5 π}{32}) : \frac{1}{3} ([\frac{1}{8} cos^{7} (u) sin (u)]_{0}^{\frac{π}{2}} + \frac{35 π}{256})

= \frac{1}{3} ([\frac{1}{8} cos^{7} (u) sin (u)]_{0}^{\frac{π}{2}} + \frac{35 π}{256})

Berechne die Grenzen:

0

= \frac{1}{3} (0 + \frac{35 π}{256})

Vereinfache

= \frac{35 π}{768}

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