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

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

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integral of xsin(1-3x)

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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