What is the integral of xsin(3x-1) ?

The integral of xsin(3x-1) is -1/3 xcos(3x-1)+1/9 sin(3x-1)+C

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integral of xsin(3x-1)

\int x sin (3 x - 1) d x

- \frac{1}{3} x cos (3 x - 1) + \frac{1}{9} sin (3 x - 1) + C

Solution steps

\int x sin (3 x - 1) d x

Apply Integration By Parts

: - \frac{1}{3} x cos (3 x - 1) - \int - \frac{1}{3} cos (3 x - 1) d x

= - \frac{1}{3} x cos (3 x - 1) - \int - \frac{1}{3} cos (3 x - 1) d x

\int - \frac{1}{3} cos (3 x - 1) d x = - \frac{1}{9} sin (3 x - 1)

= - \frac{1}{3} x cos (3 x - 1) - (- \frac{1}{9} sin (3 x - 1))

Simplify

= - \frac{1}{3} x cos (3 x - 1) + \frac{1}{9} sin (3 x - 1)

Add a constant to the solution

= - \frac{1}{3} x cos (3 x - 1) + \frac{1}{9} sin (3 x - 1) + C

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sin^{4} (x)

What is the integral of xsin(3x-1) ?
The integral of xsin(3x-1) is -1/3 xcos(3x-1)+1/9 sin(3x-1)+C