What is the integral from 0 to 2pi of cos^5(x) ?

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

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integral from 0 to 2pi of cos^5(x)

\int_{0}^{2 π} cos^{5} (x) d x

0

Solution steps

\int_{0}^{2 π} cos^{5} (x) d x

Apply Integral Reduction

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

\int_{0}^{2 π} cos^{3} (x) d x = 0

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

Simplify

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

= [\frac{1}{5} cos^{4} (x) sin (x)]_{0}^{2 π}

Compute the boundaries:

0

= 0

\int_{0}^{1} \frac{1}{x ^{2} + 2 x + 5} d x

\int_{0}^{1} sin^{2} (π) x d x

\int_{2}^{3} \frac{6}{2 x ^{2} + 2 x - 4} d x

\int_{0}^{1} 3 x^{2} x^{3} - 5 d x

\int_{1}^{5} \frac{x - 1}{x ^{2} ( x + 1 )} d x

What is the integral from 0 to 2pi of cos^5(x) ?
The integral from 0 to 2pi of cos^5(x) is 0