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

解答

\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

求解步骤

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

使用分布积分法

= [- 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}

化简

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

带入上下限计算后得:

- 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

化简

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

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