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

The integral from 0 to pi of cos^8(x) is (35pi)/(128)

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

\int_{0}^{π} cos^{8} (x) d x

\frac{35 π}{128}

Decimal Notation

0.85902 \dots

Solution steps

\int_{0}^{π} cos^{8} (x) d x

Apply Integral Reduction

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

\int_{0}^{π} cos^{6} (x) d x = \frac{5 π}{16}

= [\frac{sin ( x ) cos ^{7} ( x )}{8}]_{0}^{π} + \frac{7}{8} \cdot \frac{5 π}{16}

Simplify

[\frac{sin ( x ) cos ^{7} ( x )}{8}]_{0}^{π} + \frac{7}{8} \cdot \frac{5 π}{16} : [\frac{1}{8} cos^{7} (x) sin (x)]_{0}^{π} + \frac{35 π}{128}

= [\frac{1}{8} cos^{7} (x) sin (x)]_{0}^{π} + \frac{35 π}{128}

Compute the boundaries:

0

= 0 + \frac{35 π}{128}

Simplify

= \frac{35 π}{128}

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What is the integral from 0 to pi of cos^8(x) ?
The integral from 0 to pi of cos^8(x) is (35pi)/(128)