What is the integral of-xsin(x/2)cos(x/2) ?

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

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

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

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

Solution steps

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

Take the constant out:

\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

Apply Integration By Parts

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

= - (- \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)))

Simplify

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

Add a constant to the solution

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

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