What is the integral of x*sin(6x) ?

The integral of x*sin(6x) is -1/6 xcos(6x)+1/36 sin(6x)+C

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integral of x*sin(6x)

\int x \cdot sin (6 x) d x

- \frac{1}{6} x cos (6 x) + \frac{1}{36} sin (6 x) + C

Solution steps

\int x sin (6 x) d x

Apply Integration By Parts

: - \frac{1}{6} x cos (6 x) - \int - \frac{1}{6} cos (6 x) d x

= - \frac{1}{6} x cos (6 x) - \int - \frac{1}{6} cos (6 x) d x

\int - \frac{1}{6} cos (6 x) d x = - \frac{1}{36} sin (6 x)

= - \frac{1}{6} x cos (6 x) - (- \frac{1}{36} sin (6 x))

Simplify

= - \frac{1}{6} x cos (6 x) + \frac{1}{36} sin (6 x)

Add a constant to the solution

= - \frac{1}{6} x cos (6 x) + \frac{1}{36} sin (6 x) + C

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What is the integral of x*sin(6x) ?
The integral of x*sin(6x) is -1/6 xcos(6x)+1/36 sin(6x)+C