What is the integral of xsin((pix)/2) ?

The integral of xsin((pix)/2) is 4/(pi^2)(-pi/2 xcos((pix)/2)+sin((pix)/2))+C

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integral of xsin((pix)/2)

\int x sin (\frac{π x}{2}) d x

\frac{4}{π ^{2}} (- \frac{π}{2} x cos (\frac{π x}{2}) + sin (\frac{π x}{2})) + C

Solution steps

\int x sin (\frac{π x}{2}) d x

Apply u-substitution

: \int \frac{4 u sin ( u )}{π ^{2}} d u

= \int \frac{4 u sin ( u )}{π ^{2}} d u

Take the constant out:

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

= \frac{4}{π ^{2}} \cdot \int u sin (u) d u

Apply Integration By Parts

: - u cos (u) - \int - cos (u) d u

= \frac{4}{π ^{2}} (- u cos (u) - \int - cos (u) d u)

\int - cos (u) d u = - sin (u)

= \frac{4}{π ^{2}} (- u cos (u) - (- sin (u)))

Substitute back

u = \frac{π x}{2}

= \frac{4}{π ^{2}} (- \frac{π x}{2} cos (\frac{π x}{2}) - (- sin (\frac{π x}{2})))

Simplify

\frac{4}{π ^{2}} (- \frac{π x}{2} cos (\frac{π x}{2}) - (- sin (\frac{π x}{2}))) : \frac{4}{π ^{2}} (- \frac{π}{2} x cos (\frac{π x}{2}) + sin (\frac{π x}{2}))

= \frac{4}{π ^{2}} (- \frac{π}{2} x cos (\frac{π x}{2}) + sin (\frac{π x}{2}))

Add a constant to the solution

= \frac{4}{π ^{2}} (- \frac{π}{2} x cos (\frac{π x}{2}) + sin (\frac{π x}{2})) + C

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What is the integral of xsin((pix)/2) ?
The integral of xsin((pix)/2) is 4/(pi^2)(-pi/2 xcos((pix)/2)+sin((pix)/2))+C