What is the integral of xsin(-5x) ?

The integral of xsin(-5x) is 1/5 xcos(5x)-1/25 sin(5x)+C

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integral of xsin(-5x)

\int x sin (- 5 x) d x

\frac{1}{5} x cos (5 x) - \frac{1}{25} sin (5 x) + C

Solution steps

\int x sin (- 5 x) d x

Apply Integration By Parts

= - \int - \frac{5}{2} x^{2} cos (5 x) d x - \frac{1}{2} x^{2} sin (5 x)

\int - \frac{5}{2} x^{2} cos (5 x) d x = - \frac{1}{50} (25 x^{2} sin (5 x) - 2 (- 5 x cos (5 x) + sin (5 x)))

= - (- \frac{1}{50} (25 x^{2} sin (5 x) - 2 (- 5 x cos (5 x) + sin (5 x)))) - \frac{1}{2} x^{2} sin (5 x)

Simplify

- (- \frac{1}{50} (25 x^{2} sin (5 x) - 2 (- 5 x cos (5 x) + sin (5 x)))) - \frac{1}{2} x^{2} sin (5 x) : \frac{1}{5} x cos (5 x) - \frac{1}{25} sin (5 x)

= \frac{1}{5} x cos (5 x) - \frac{1}{25} sin (5 x)

Add a constant to the solution

= \frac{1}{5} x cos (5 x) - \frac{1}{25} sin (5 x) + C

What is the integral of xsin(-5x) ?
The integral of xsin(-5x) is 1/5 xcos(5x)-1/25 sin(5x)+C