What is the integral from 0 to l/2 of xsin((npix)/l) ?

The integral from 0 to l/2 of xsin((npix)/l) is (l^2sin((pin)/2))/(pi^2n^2)-(l^2cos((pin)/2))/(2pin)

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integral from 0 to l/2 of xsin((npix)/l)

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

\frac{l ^{2} sin ( \frac{πn}{2} )}{π ^{2} n ^{2}} - \frac{l ^{2} cos ( \frac{πn}{2} )}{2 πn}

Solution steps

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

Apply Integration By Parts

: [- l \frac{1}{πn} x cos (\frac{πn x}{l}) - \int - l \frac{1}{πn} cos (\frac{πn x}{l}) d x]_{0}^{\frac{l}{2}}

= [- l \frac{1}{πn} x cos (\frac{πn x}{l}) - \int - l \frac{1}{πn} cos (\frac{πn x}{l}) d x]_{0}^{\frac{l}{2}}

\int - l \frac{1}{πn} cos (\frac{πn x}{l}) d x = - l \frac{l}{π ^{2} n ^{2}} sin (\frac{πn x}{l})

= [- l \frac{1}{πn} x cos (\frac{πn x}{l}) - (- l \frac{l}{π ^{2} n ^{2}} sin (\frac{πn x}{l}))]_{0}^{\frac{l}{2}}

Simplify

= [- l \frac{1}{πn} x cos (\frac{πn x}{l}) + l \frac{l}{π ^{2} n ^{2}} sin (\frac{πn x}{l})]_{0}^{\frac{l}{2}}

Compute the boundaries:

\frac{l ^{2}}{π ^{2} n ^{2}} sin (\frac{πn}{2}) - \frac{l ^{2}}{2 πn} cos (\frac{πn}{2})

= \frac{l ^{2}}{π ^{2} n ^{2}} sin (\frac{πn}{2}) - \frac{l ^{2}}{2 πn} cos (\frac{πn}{2})

Simplify

= \frac{l ^{2} sin ( \frac{πn}{2} )}{π ^{2} n ^{2}} - \frac{l ^{2} cos ( \frac{πn}{2} )}{2 πn}

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What is the integral from 0 to l/2 of xsin((npix)/l) ?
The integral from 0 to l/2 of xsin((npix)/l) is (l^2sin((pin)/2))/(pi^2n^2)-(l^2cos((pin)/2))/(2pin)