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

Lösung

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

\frac{5 π}{16}

Dezimale

0.98174 \dots

Schritte zur Lösung

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

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Wende integrale Reduktion an

= 2 ([\frac{sin ( u ) cos ^{5} ( u )}{6}]_{0}^{\frac{π}{2}} + \frac{5}{6} \cdot \int_{0}^{\frac{π}{2}} cos^{4} (u) d u)

\int_{0}^{\frac{π}{2}} cos^{4} (u) d u = \frac{3 π}{16}

= 2 ([\frac{sin ( u ) cos ^{5} ( u )}{6}]_{0}^{\frac{π}{2}} + \frac{5}{6} \cdot \frac{3 π}{16})

Vereinfache

2 ([\frac{sin ( u ) cos ^{5} ( u )}{6}]_{0}^{\frac{π}{2}} + \frac{5}{6} \cdot \frac{3 π}{16}) : 2 ([\frac{1}{6} cos^{5} (u) sin (u)]_{0}^{\frac{π}{2}} + \frac{5 π}{32})

= 2 ([\frac{1}{6} cos^{5} (u) sin (u)]_{0}^{\frac{π}{2}} + \frac{5 π}{32})

Berechne die Grenzen:

0

= 2 (0 + \frac{5 π}{32})

Vereinfache

= \frac{5 π}{16}

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