What is the integral of xsin(cx) ?

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

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integral of xsin(cx)

\int x sin (c x) d x

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

Solution steps

\int x sin (c x) d x

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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