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

The integral from pi to 2pi of (x-2sin(x)) is 2pi^2-(pi^2)/2+4

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

\int_{π}^{2 π} (x - 2 sin (x)) d x

2 π^{2} - \frac{π ^{2}}{2} + 4

Decimal

18.80440 \dots

Solution steps

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

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

\int_{π}^{2 π} 2 sin (x) d x = - 4

= 2 π^{2} - \frac{π ^{2}}{2} - (- 4)

Simplify

= 2 π^{2} - \frac{π ^{2}}{2} + 4

What is the integral from pi to 2pi of (x-2sin(x)) ?
The integral from pi to 2pi of (x-2sin(x)) is 2pi^2-(pi^2)/2+4