integral of-xsin(x/2)cos(x/2)

解答

\int - x sin (\frac{x}{2}) cos (\frac{x}{2}) d x

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

求解步骤

\int - x sin (\frac{x}{2}) cos (\frac{x}{2}) d x

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int x sin (\frac{x}{2}) cos (\frac{x}{2}) d x

使用分布积分法

= - (- \frac{1}{2} x cos (x) - \int - \frac{1}{2} cos (x) d x)

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

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

化简

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

解答补常数

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

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