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

The integral of x^2sin(x/2) is -2x^2cos(x/2)+16(1/2 xsin(x/2)+cos(x/2))+C

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integral of x^2sin(x/2)

\int x^{2} sin (\frac{x}{2}) d x

- 2 x^{2} cos (\frac{x}{2}) + 16 (\frac{1}{2} x sin (\frac{x}{2}) + cos (\frac{x}{2})) + C

Solution steps

\int 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

\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})))

Simplify

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

Add a constant to the solution

= - 2 x^{2} cos (\frac{x}{2}) + 16 (\frac{1}{2} x sin (\frac{x}{2}) + cos (\frac{x}{2})) + C

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What is the integral of x^2sin(x/2) ?
The integral of x^2sin(x/2) is -2x^2cos(x/2)+16(1/2 xsin(x/2)+cos(x/2))+C