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

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

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integral of xsin(-3x)

\int x sin (- 3 x) d x

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

Solution steps

\int x sin (- 3 x) d x

Apply Integration By Parts

= - \int - \frac{3}{2} x^{2} cos (3 x) d x - \frac{1}{2} x^{2} sin (3 x)

\int - \frac{3}{2} x^{2} cos (3 x) d x = - \frac{1}{18} (9 x^{2} sin (3 x) - 2 (- 3 x cos (3 x) + sin (3 x)))

= - (- \frac{1}{18} (9 x^{2} sin (3 x) - 2 (- 3 x cos (3 x) + sin (3 x)))) - \frac{1}{2} x^{2} sin (3 x)

Simplify

- (- \frac{1}{18} (9 x^{2} sin (3 x) - 2 (- 3 x cos (3 x) + sin (3 x)))) - \frac{1}{2} x^{2} sin (3 x) : \frac{1}{3} x cos (3 x) - \frac{1}{9} sin (3 x)

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

Add a constant to the solution

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

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