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

The integral from pi to 2pi of xsin(x/2) is 4pi-4

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

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

4 π - 4

Decimal Notation

8.56637 \dots

Solution steps

\int_{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]_{π}^{2 π}

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

Simplify

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

Compute the boundaries:

4 π - 4

= 4 π - 4

y^{^{''}} + 12 y^{^{'}} + 36 y = t^{- 2} e^{- 6 t}

y^{^{''}} - 2 y^{^{'}} + 2 y = 0

y = 4 - x^{2}, (- 1, 3)

\frac{1}{3 s + 1}

y^{^{''}} + 14 y^{^{'}} + 49 y = t^{- 2} e^{- 7 t}

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