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

The integral from 0 to pi of cos^6(x) is (5pi)/(16)

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

\int_{0}^{π} cos^{6} (x) d x

\frac{5 π}{16}

Decimal Notation

0.98174 \dots

Solution steps

\int_{0}^{π} cos^{6} (x) d x

Apply Integral Reduction

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

\int_{0}^{π} cos^{4} (x) d x = \frac{3 π}{8}

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

Simplify

[\frac{sin ( x ) cos ^{5} ( x )}{6}]_{0}^{π} + \frac{5}{6} \cdot \frac{3 π}{8} : [\frac{1}{6} cos^{5} (x) sin (x)]_{0}^{π} + \frac{5 π}{16}

= [\frac{1}{6} cos^{5} (x) sin (x)]_{0}^{π} + \frac{5 π}{16}

Compute the boundaries:

0

= 0 + \frac{5 π}{16}

Simplify

= \frac{5 π}{16}

y^{^{'}} = 0.285 (1 - y)

100 y^{^{''}} = 0

x \to 3 lim (5 x^{2})

\frac{d}{d x} (x^{- \frac{2}{3}} (x + 5))

\frac{d}{d x} (\frac{6 x + 8 x ^{2}}{( 3 + 8 x ) ^{2}})

What is the integral from 0 to pi of cos^6(x) ?
The integral from 0 to pi of cos^6(x) is (5pi)/(16)