What is the integral from 0 to 1 of xsin((npix)/1) ?

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

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integral from 0 to 1 of xsin((npix)/1)

\int_{0}^{1} x sin (\frac{nπ x}{1}) d x

- \frac{( - 1 ) ^{n}}{πn}

Solution steps

\int_{0}^{1} x sin (\frac{nπ x}{1}) d x

Simplify

= \int_{0}^{1} x sin (πn x) d x

Apply Integration By Parts

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

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

\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))]_{0}^{1}

Simplify

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

Compute the boundaries:

- (- 1)^{n} \frac{1}{πn}

= - (- 1)^{n} \frac{1}{πn}

Simplify

= - \frac{( - 1 ) ^{n}}{πn}

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