What is the integral from 0 to 2pi of sin^6(x) ?

The integral from 0 to 2pi of sin^6(x) is (5pi)/8

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

\int_{0}^{2 π} sin^{6} (x) d x

\frac{5 π}{8}

Decimal Notation

1.96349 \dots

Solution steps

\int_{0}^{2 π} sin^{6} (x) d x

Apply Integral Reduction

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

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

= - [\frac{cos ( x ) sin ^{5} ( x )}{6}]_{0}^{2 π} + \frac{5}{6} \cdot \frac{3 π}{4}

Simplify

- [\frac{cos ( x ) sin ^{5} ( x )}{6}]_{0}^{2 π} + \frac{5}{6} \cdot \frac{3 π}{4} : - [\frac{1}{6} sin^{5} (x) cos (x)]_{0}^{2 π} + \frac{5 π}{8}

= - [\frac{1}{6} sin^{5} (x) cos (x)]_{0}^{2 π} + \frac{5 π}{8}

Compute the boundaries:

0

= - 0 + \frac{5 π}{8}

Simplify

= \frac{5 π}{8}

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What is the integral from 0 to 2pi of sin^6(x) ?
The integral from 0 to 2pi of sin^6(x) is (5pi)/8