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

The integral from pi/2 to pi of x^2sin(x/2) is 8pi+(pi^2)/(2sqrt(2))-2sqrt(2)pi-8sqrt(2)

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integral from pi/2 to pi of x^2sin(x/2)

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

8 π + \frac{π ^{2}}{2 2} - 22 π - 82

Decimal Notation

8.42269 \dots

Solution steps

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

Apply Integration By Parts

: [- 2 x^{2} cos (\frac{x}{2}) - \int - 4 x cos (\frac{x}{2}) d x]_{\frac{π}{2}}^{π}

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

\int - 4 x cos (\frac{x}{2}) d x = - 16 (\frac{1}{2} x sin (\frac{x}{2}) + cos (\frac{x}{2}))

= [- 2 x^{2} cos (\frac{x}{2}) - (- 16 (\frac{1}{2} x sin (\frac{x}{2}) + cos (\frac{x}{2})))]_{\frac{π}{2}}^{π}

Simplify

= [- 2 x^{2} cos (\frac{x}{2}) + 16 (\frac{1}{2} x sin (\frac{x}{2}) + cos (\frac{x}{2}))]_{\frac{π}{2}}^{π}

Compute the boundaries:

8 π + \frac{π ^{2}}{2 2} - 22 π - 82

= 8 π + \frac{π ^{2}}{2 2} - 22 π - 82

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What is the integral from pi/2 to pi of x^2sin(x/2) ?
The integral from pi/2 to pi of x^2sin(x/2) is 8pi+(pi^2)/(2sqrt(2))-2sqrt(2)pi-8sqrt(2)