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

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

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

\int_{0}^{2 π} cos^{6} (x) d x

\frac{5 π}{8}

Solution steps

\int_{0}^{2 π} cos^{6} (x) d x

Apply Integral Reduction

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

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

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

Simplify

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

= [\frac{1}{6} cos^{5} (x) sin (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 cos^6(x) ?
The integral from 0 to 2pi of cos^6(x) is (5pi)/8