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

The integral from 0 to l of xsin((npix)/l) is -((-1)^nl^2)/(pin)

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

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

- \frac{( - 1 ) ^{n} l ^{2}}{πn}

Solution steps

\int_{0}^{l} 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}^{l}

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

\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}^{l}

Simplify

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

Compute the boundaries:

- \frac{( - 1 ) ^{n} l ^{2}}{πn}

= - \frac{( - 1 ) ^{n} l ^{2}}{πn}

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What is the integral from 0 to l of xsin((npix)/l) ?
The integral from 0 to l of xsin((npix)/l) is -((-1)^nl^2)/(pin)