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

The integral from pi/2 to 0 of (1+cos(2x))/2 is -pi/4

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

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

- \frac{π}{4}

Decimal

- 0.78539 \dots

Solution steps

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

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

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

\int_{0}^{\frac{π}{2}} 1 d x = \frac{π}{2}

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

= - \frac{1}{2} (\frac{π}{2} + 0)

Simplify

- \frac{1}{2} (\frac{π}{2} + 0) : - \frac{π}{4}

= - \frac{π}{4}

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