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

The integral from-2pi to 2pi of xsin(x/2) is 8pi

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

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

8 π

Decimal

25.13274 \dots

Solution steps

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

Apply Integration By Parts

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

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

\int - 2 cos (\frac{x}{2}) d x = - 4 sin (\frac{x}{2})

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

Simplify

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

Compute the boundaries:

8 π

= 8 π

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\int_{- 1}^{1} e^{x} + e^{- x} d x

\int_{0}^{2 - x^{2} - y^{2}} (x y e^{z}) d z

\int_{- h}^{h} (y^{2} - h^{2}) d y

What is the integral from-2pi to 2pi of xsin(x/2) ?
The integral from-2pi to 2pi of xsin(x/2) is 8pi