integral from 0 to pi/6 of sin^4(x)

Lösung

\int_{0}^{\frac{π}{6}} sin^{4} (x) d x

\frac{4 π - 7 3}{64}

Dezimale

0.00690 \dots

Schritte zur Lösung

\int_{0}^{\frac{π}{6}} sin^{4} (x) d x

Wende integrale Reduktion an

= - [\frac{cos ( x ) sin ^{3} ( x )}{4}]_{0}^{\frac{π}{6}} + \frac{3}{4} \cdot \int_{0}^{\frac{π}{6}} sin^{2} (x) d x

\int_{0}^{\frac{π}{6}} sin^{2} (x) d x = \frac{2 π - 3 3}{24}

= - [\frac{cos ( x ) sin ^{3} ( x )}{4}]_{0}^{\frac{π}{6}} + \frac{3}{4} \cdot \frac{2 π - 3 3}{24}

Vereinfache

- [\frac{cos ( x ) sin ^{3} ( x )}{4}]_{0}^{\frac{π}{6}} + \frac{3}{4} \cdot \frac{2 π - 3 3}{24} : - [\frac{1}{4} sin^{3} (x) cos (x)]_{0}^{\frac{π}{6}} + \frac{2 π - 3 3}{32}

= - [\frac{1}{4} sin^{3} (x) cos (x)]_{0}^{\frac{π}{6}} + \frac{2 π - 3 3}{32}

Berechne die Grenzen:

\frac{3}{64}

= - \frac{3}{64} + \frac{2 π - 3 3}{32}

Vereinfache

= \frac{4 π - 7 3}{64}

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