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

The integral from 0 to pi/3 of (1+cos(x))^2 is sqrt(3)+(4pi+sqrt(3))/8

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

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

3 + \frac{4 π + 3}{8}

Decimal Notation

3.51935 \dots

Solution steps

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

Expand

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

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

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

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

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

\int_{0}^{\frac{π}{3}} cos^{2} (x) d x = \frac{4 π + 3 3}{24}

= \frac{π}{3} + 3 + \frac{4 π + 3 3}{24}

Simplify

\frac{π}{3} + 3 + \frac{4 π + 3 3}{24} : 3 + \frac{4 π + 3}{8}

= 3 + \frac{4 π + 3}{8}

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What is the integral from 0 to pi/3 of (1+cos(x))^2 ?
The integral from 0 to pi/3 of (1+cos(x))^2 is sqrt(3)+(4pi+sqrt(3))/8