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

The integral from 0 to pi/3 of (1-2cos(θ))^2 is -2sqrt(3)+(2pi+sqrt(3))/2

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

\int_{0}^{\frac{π}{3}} (1 - 2 cos (θ))^{2} d θ

- 23 + \frac{2 π + 3}{2}

Decimal

0.54351 \dots

Solution steps

\int_{0}^{\frac{π}{3}} (1 - 2 cos (θ))^{2} d θ

Expand

(1 - 2 cos (θ))^{2} : 1 - 4 cos (θ) + 4 cos^{2} (θ)

= \int_{0}^{\frac{π}{3}} 1 - 4 cos (θ) + 4 cos^{2} (θ) d θ

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}^{\frac{π}{3}} 1 d θ - \int_{0}^{\frac{π}{3}} 4 cos (θ) d θ + \int_{0}^{\frac{π}{3}} 4 cos^{2} (θ) d θ

\int_{0}^{\frac{π}{3}} 1 d θ = \frac{π}{3}

\int_{0}^{\frac{π}{3}} 4 cos (θ) d θ = 23

\int_{0}^{\frac{π}{3}} 4 cos^{2} (θ) d θ = 2 (\frac{π}{3} + \frac{3}{4})

= \frac{π}{3} - 23 + 2 (\frac{π}{3} + \frac{3}{4})

Simplify

\frac{π}{3} - 23 + 2 (\frac{π}{3} + \frac{3}{4}) : - 23 + \frac{2 π + 3}{2}

= - 23 + \frac{2 π + 3}{2}

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