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

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

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

\int_{0}^{π} x sin (\frac{x}{2}) d x

4

Solution steps

\int_{0}^{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}^{π}

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

Simplify

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

Compute the boundaries:

4

= 4

\int_{0}^{1} a + b x^{2} d x

\int_{0}^{1} \frac{1 - cos ( x )}{x ^{2}} d x

\int_{0}^{\frac{π}{2}} \frac{1}{1 + sin ( x )} d x

\int_{0}^{4} (x^{2} - x + 1) e^{2 x} d x

\int_{0}^{2} x ln (x + 1) d x

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