What is the integral from 0 to 2pi of cos^8(x) ?

The integral from 0 to 2pi of cos^8(x) is (35pi)/(64)

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

\int_{0}^{2 π} cos^{8} (x) d x

\frac{35 π}{64}

Decimal

1.71805 \dots

Solution steps

\int_{0}^{2 π} cos^{8} (x) d x

Apply Integral Reduction

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

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

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

Simplify

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

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

Compute the boundaries:

0

= 0 + \frac{35 π}{64}

Simplify

= \frac{35 π}{64}

What is the integral from 0 to 2pi of cos^8(x) ?
The integral from 0 to 2pi of cos^8(x) is (35pi)/(64)