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

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

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integral from 0 to 2pi of (cos(x))^3

\int_{0}^{2 π} (cos (x))^{3} d x

0

Solution steps

\int_{0}^{2 π} (cos (x))^{3} d x

Simplify

= \int_{0}^{2 π} cos^{3} (x) d x

Simplify

cos^{3} (x) : cos^{2} (x) cos (x)

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

Rewrite using trig identities

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

Apply Integration By Parts

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

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

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

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

Simplify

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

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

Compute the boundaries:

0

= 0

\int_{1}^{25} \frac{1}{x} d x

\int_{5}^{9} \frac{1}{x ^{2}} d x

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

\int_{2}^{6} 30 x^{3} 1 + x^{2} d x

\int_{0}^{\frac{1}{4}} \frac{2}{1 + 16 x ^{2}} d x

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