What is the integral of xcos((nxpi)/l) ?

The integral of xcos((nxpi)/l) is l 1/(pin)xsin((pinx)/l)+l l/(pi^2n^2)cos((pinx)/l)+C

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integral of xcos((nxpi)/l)

\int x cos (\frac{n x π}{l}) d x

l \frac{1}{πn} x sin (\frac{πn x}{l}) + l \frac{l}{π ^{2} n ^{2}} cos (\frac{πn x}{l}) + C

Solution steps

\int x cos (\frac{n x π}{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

= l \frac{1}{πn} x sin (\frac{πn x}{l}) - \int l \frac{1}{πn} sin (\frac{πn x}{l}) d x

\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}))

Simplify

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

Add a constant to the solution

= l \frac{1}{πn} x sin (\frac{πn x}{l}) + l \frac{l}{π ^{2} n ^{2}} cos (\frac{πn x}{l}) + C

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What is the integral of xcos((nxpi)/l) ?
The integral of xcos((nxpi)/l) is l 1/(pin)xsin((pinx)/l)+l l/(pi^2n^2)cos((pinx)/l)+C