What is the integral of xsin(wx) ?

The integral of xsin(wx) is -1/w xcos(wx)+1/(w^2)sin(wx)+C

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integral of xsin(wx)

\int x sin (w x) d x

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

Solution steps

\int 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

\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))

Simplify

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

Add a constant to the solution

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

What is the integral of xsin(wx) ?
The integral of xsin(wx) is -1/w xcos(wx)+1/(w^2)sin(wx)+C