What is the integral of xsin(npix) ?

The integral of xsin(npix) is -1/(pin)xcos(pinx)+1/(pi^2n^2)sin(pinx)+C

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integral of xsin(npix)

\int x sin (nπ x) d x

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

Solution steps

\int x sin (nπ x) d x

Apply Integration By Parts

: - \frac{1}{πn} x cos (πn x) - \int - \frac{1}{πn} cos (πn x) d x

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

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

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

Simplify

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

Add a constant to the solution

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

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What is the integral of xsin(npix) ?
The integral of xsin(npix) is -1/(pin)xcos(pinx)+1/(pi^2n^2)sin(pinx)+C