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

The integral from 0 to 1 of xsin(wx) is (sin(w)}{w^2}-\frac{cos(w))/w

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integral from 0 to 1 of xsin(wx)

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

\frac{sin ( w )}{w ^{2}} - \frac{cos ( w )}{w}

Solution steps

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

Apply Integration By Parts

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

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

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

= [- \frac{1}{w} x cos (w x) - (- \frac{1}{w ^{2}} sin (w x))]_{0}^{1}

Simplify

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

Compute the boundaries:

\frac{1}{w ^{2}} sin (w) - \frac{1}{w} cos (w)

= \frac{1}{w ^{2}} sin (w) - \frac{1}{w} cos (w)

Simplify

= \frac{sin ( w )}{w ^{2}} - \frac{cos ( w )}{w}

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What is the integral from 0 to 1 of xsin(wx) ?
The integral from 0 to 1 of xsin(wx) is (sin(w)}{w^2}-\frac{cos(w))/w