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

The integral from 0 to l of xcos((npix)/l) is ((-1)^nl^2-l^2)/(pi^2n^2)

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

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

\frac{( - 1 ) ^{n} l ^{2} - l ^{2}}{π ^{2} n ^{2}}

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Compute the boundaries:

\frac{( - 1 ) ^{n} l ^{2}}{π ^{2} n ^{2}} - \frac{l ^{2}}{π ^{2} n ^{2}}

= \frac{( - 1 ) ^{n} l ^{2}}{π ^{2} n ^{2}} - \frac{l ^{2}}{π ^{2} n ^{2}}

Simplify

= \frac{( - 1 ) ^{n} l ^{2} - l ^{2}}{π ^{2} n ^{2}}

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