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

The integral from 0 to 2pi of 1/2 (3+3cos(x))^2 is (27pi)/2

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

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

\frac{27 π}{2}

Decimal

42.41150 \dots

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Expand

(3 + 3 cos (x))^{2} : 9 + 18 cos (x) + 9 cos^{2} (x)

= \frac{1}{2} \cdot \int_{0}^{2 π} 9 + 18 cos (x) + 9 cos^{2} (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

= \frac{1}{2} (\int_{0}^{2 π} 9 d x + \int_{0}^{2 π} 18 cos (x) d x + \int_{0}^{2 π} 9 cos^{2} (x) d x)

\int_{0}^{2 π} 9 d x = 18 π

\int_{0}^{2 π} 18 cos (x) d x = 0

\int_{0}^{2 π} 9 cos^{2} (x) d x = 9 π

= \frac{1}{2} (18 π + 0 + 9 π)

Simplify

\frac{1}{2} (18 π + 0 + 9 π) : \frac{27 π}{2}

= \frac{27 π}{2}

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What is the integral from 0 to 2pi of 1/2 (3+3cos(x))^2 ?
The integral from 0 to 2pi of 1/2 (3+3cos(x))^2 is (27pi)/2