What is the integral from 0 to a of x^2sin(x/2) ?

The integral from 0 to a of x^2sin(x/2) is -2a^2cos(a/2)+16((asin(a/2))/2+cos(a/2))-16

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integral from 0 to a of x^2sin(x/2)

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

- 2 a^{2} cos (\frac{a}{2}) + 16 (\frac{a sin ( \frac{a}{2} )}{2} + cos (\frac{a}{2})) - 16

Solution steps

\int_{0}^{a} 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]_{0}^{a}

= [- 2 x^{2} cos (\frac{x}{2}) - \int - 4 x cos (\frac{x}{2}) d x]_{0}^{a}

\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})))]_{0}^{a}

Simplify

= [- 2 x^{2} cos (\frac{x}{2}) + 16 (\frac{1}{2} x sin (\frac{x}{2}) + cos (\frac{x}{2}))]_{0}^{a}

Compute the boundaries:

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

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

Simplify

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

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What is the integral from 0 to a of x^2sin(x/2) ?
The integral from 0 to a of x^2sin(x/2) is -2a^2cos(a/2)+16((asin(a/2))/2+cos(a/2))-16