What is the integral from-2pi to 2pi of (pi-x)/2 ?

The integral from-2pi to 2pi of (pi-x)/2 is 2pi^2

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integral from-2pi to 2pi of (pi-x)/2

\int_{- 2 π}^{2 π} \frac{π - x}{2} d x

2 π^{2}

Decimal

19.73920 \dots

Solution steps

\int_{- 2 π}^{2 π} \frac{π - 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_{- 2 π}^{2 π} π - 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_{- 2 π}^{2 π} π d x - \int_{- 2 π}^{2 π} x d x)

\int_{- 2 π}^{2 π} π d x = 4 π^{2}

\int_{- 2 π}^{2 π} x d x = 0

= \frac{1}{2} (4 π^{2} - 0)

Simplify

\frac{1}{2} (4 π^{2} - 0) : 2 π^{2}

= 2 π^{2}

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What is the integral from-2pi to 2pi of (pi-x)/2 ?
The integral from-2pi to 2pi of (pi-x)/2 is 2pi^2