What is the integral of xcos(3x)-sin(3x) ?

The integral of xcos(3x)-sin(3x) is 1/3 xsin(3x)+4/9 cos(3x)+C

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integral of xcos(3x)-sin(3x)

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

\frac{1}{3} x sin (3 x) + \frac{4}{9} cos (3 x) + C

Solution steps

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x cos (3 x) d x - \int sin (3 x) d x

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

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

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

Simplify

\frac{1}{9} (3 x sin (3 x) + cos (3 x)) - (- \frac{1}{3} cos (3 x)) : \frac{1}{3} x sin (3 x) + \frac{4}{9} cos (3 x)

= \frac{1}{3} x sin (3 x) + \frac{4}{9} cos (3 x)

Add a constant to the solution

= \frac{1}{3} x sin (3 x) + \frac{4}{9} cos (3 x) + C

What is the integral of xcos(3x)-sin(3x) ?
The integral of xcos(3x)-sin(3x) is 1/3 xsin(3x)+4/9 cos(3x)+C