What is the integral from-2pi to 2pi of x*sin(0.5x) ?

The integral from-2pi to 2pi of x*sin(0.5x) is 25.13274…

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integral from-2pi to 2pi of x*sin(0.5x)

\int_{- 2 π}^{2 π} x \cdot sin (0.5 x) d x

25.13274 \dots

Solution steps

\int_{- 2 π}^{2 π} x sin (0.5 x) d x

Apply Integration By Parts

: [- 2 x cos (0.5 x) - \int - 2 cos (0.5 x) d x]_{- 6.28318 \dots}^{2 π}

= [- 2 x cos (0.5 x) - \int - 2 cos (0.5 x) d x]_{- 6.28318 \dots}^{2 π}

\int - 2 cos (0.5 x) d x = - 4 sin (0.5 x)

= [- 2 x cos (0.5 x) - (- 4 sin (0.5 x))]_{- 6.28318 \dots}^{2 π}

Simplify

= [- 2 x cos (0.5 x) + 4 sin (0.5 x)]_{- 6.28318 \dots}^{2 π}

Compute the boundaries:

25.13274 \dots

= 25.13274 \dots

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What is the integral from-2pi to 2pi of x*sin(0.5x) ?
The integral from-2pi to 2pi of x*sin(0.5x) is 25.13274…