integral from 0 to 2pi of sin^8(x)

Lösung

\int_{0}^{2 π} sin^{8} (x) d x

\frac{35 π}{64}

Dezimale

1.71805 \dots

Schritte zur Lösung

\int_{0}^{2 π} sin^{8} (x) d x

Wende integrale Reduktion an

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

\int_{0}^{2 π} sin^{6} (x) d x = \frac{5 π}{8}

= - [\frac{cos ( x ) sin ^{7} ( x )}{8}]_{0}^{2 π} + \frac{7}{8} \cdot \frac{5 π}{8}

Vereinfache

- [\frac{cos ( x ) sin ^{7} ( x )}{8}]_{0}^{2 π} + \frac{7}{8} \cdot \frac{5 π}{8} : - [\frac{1}{8} sin^{7} (x) cos (x)]_{0}^{2 π} + \frac{35 π}{64}

= - [\frac{1}{8} sin^{7} (x) cos (x)]_{0}^{2 π} + \frac{35 π}{64}

Berechne die Grenzen:

0

= - 0 + \frac{35 π}{64}

Vereinfache

= \frac{35 π}{64}

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