What is the integral from 0 to 2pi of cos^3(θ) ?

The integral from 0 to 2pi of cos^3(θ) is 0

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integral from 0 to 2pi of cos^3(θ)

\int_{0}^{2 π} cos^{3} (θ) d θ

0

Solution steps

\int_{0}^{2 π} cos^{3} (θ) d θ

Simplify

cos^{3} (θ) : cos^{2} (θ) cos (θ)

= \int_{0}^{2 π} cos^{2} (θ) cos (θ) d θ

Rewrite using trig identities

= \int_{0}^{2 π} (1 - sin^{2} (θ)) cos (θ) d θ

Apply Integration By Parts

= [- \int - sin (2 θ) sin (θ) d θ + cos^{2} (θ) sin (θ)]_{0}^{2 π}

\int - sin (2 θ) sin (θ) d θ = - \frac{2}{3} sin^{3} (θ)

= [- (- \frac{2}{3} sin^{3} (θ)) + cos^{2} (θ) sin (θ)]_{0}^{2 π}

Simplify

[- (- \frac{2}{3} sin^{3} (θ)) + cos^{2} (θ) sin (θ)]_{0}^{2 π} : [\frac{1}{3} (2 sin^{3} (θ) + 3 cos^{2} (θ) sin (θ))]_{0}^{2 π}

= [\frac{1}{3} (2 sin^{3} (θ) + 3 cos^{2} (θ) sin (θ))]_{0}^{2 π}

Compute the boundaries:

0

= 0

What is the integral from 0 to 2pi of cos^3(θ) ?
The integral from 0 to 2pi of cos^3(θ) is 0