What is the integral from 0 to 2pi of (1-cos(x))^3 ?

The integral from 0 to 2pi of (1-cos(x))^3 is 5pi

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integral from 0 to 2pi of (1-cos(x))^3

\int_{0}^{2 π} (1 - cos (x))^{3} d x

5 π

Decimal Notation

15.70796 \dots

Solution steps

\int_{0}^{2 π} (1 - cos (x))^{3} d x

Expand

(1 - cos (x))^{3} : 1 - 3 cos (x) + 3 cos^{2} (x) - cos^{3} (x)

= \int_{0}^{2 π} 1 - 3 cos (x) + 3 cos^{2} (x) - cos^{3} (x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{0}^{2 π} 1 d x - \int_{0}^{2 π} 3 cos (x) d x + \int_{0}^{2 π} 3 cos^{2} (x) d x - \int_{0}^{2 π} cos^{3} (x) d x

\int_{0}^{2 π} 1 d x = 2 π

\int_{0}^{2 π} 3 cos (x) d x = 0

\int_{0}^{2 π} 3 cos^{2} (x) d x = 3 π

\int_{0}^{2 π} cos^{3} (x) d x = 0

= 2 π - 0 + 3 π - 0

Simplify

= 5 π

\int_{0}^{2 π} cos^{3} (t) d t

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What is the integral from 0 to 2pi of (1-cos(x))^3 ?
The integral from 0 to 2pi of (1-cos(x))^3 is 5pi