What is the integral from 0 to 2 of xcos(wx) ?

The integral from 0 to 2 of xcos(wx) is (2wsin(2w)+cos(2w)-1)/(w^2)

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integral from 0 to 2 of xcos(wx)

\int_{0}^{2} x cos (w x) d x

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

Solution steps

\int_{0}^{2} x cos (w x) d x

Apply u-substitution

: \int_{0}^{w \cdot 2} \frac{u cos ( u )}{w ^{2}} d u

= \int_{0}^{w \cdot 2} \frac{u cos ( u )}{w ^{2}} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{w ^{2}} \cdot \int_{0}^{w \cdot 2} u cos (u) d u

Apply Integration By Parts

: [u sin (u) - \int sin (u) d u]_{0}^{2 w}

= \frac{1}{w ^{2}} [u sin (u) - \int sin (u) d u]_{0}^{2 w}

\int sin (u) d u = - cos (u)

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

Simplify

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

Compute the boundaries:

2 w sin (2 w) + cos (2 w) - 1

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

Simplify

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

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What is the integral from 0 to 2 of xcos(wx) ?
The integral from 0 to 2 of xcos(wx) is (2wsin(2w)+cos(2w)-1)/(w^2)