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

Lösung

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

\frac{5 π}{32}

Dezimale

0.49087 \dots

Schritte zur Lösung

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

Wende integrale Reduktion an

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

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

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

Vereinfache

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

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

Berechne die Grenzen:

0

= 0 + \frac{5 π}{32}

Vereinfache

= \frac{5 π}{32}

\frac{d}{d x} (- sin (5 x) \cdot 5)

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d er i v a t i v e sin (cos (5 x))

\int sin (x) + 8 d x

\int \frac{8}{x ^{8}} d x