What is the integral from 0 to 2pi of xsin(x/2) ?

The integral from 0 to 2pi of xsin(x/2) is 4pi

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integral from 0 to 2pi of xsin(x/2)

\int_{0}^{2 π} x sin (\frac{x}{2}) d x

4 π

Decimal

12.56637 \dots

Solution steps

\int_{0}^{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]_{0}^{2 π}

= [- 2 x cos (\frac{x}{2}) - \int - 2 cos (\frac{x}{2}) d x]_{0}^{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}))]_{0}^{2 π}

Simplify

= [- 2 x cos (\frac{x}{2}) + 4 sin (\frac{x}{2})]_{0}^{2 π}

Compute the boundaries:

4 π

= 4 π

\int_{1}^{2} x e^{\frac{x}{2}} d x

\int_{1}^{3} \frac{1}{2 x - 1} d x

\int_{0}^{\frac{π}{2}} x^{3} sin (x) d x

\int_{0}^{1} \frac{7}{1 + p ^{2}} d p

\int_{0}^{1} sin (nπ t) d t

What is the integral from 0 to 2pi of xsin(x/2) ?
The integral from 0 to 2pi of xsin(x/2) is 4pi