What is the integral of xsin(-4x) ?

The integral of xsin(-4x) is 1/4 xcos(4x)-1/16 sin(4x)+C

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integral of xsin(-4x)

\int x sin (- 4 x) d x

\frac{1}{4} x cos (4 x) - \frac{1}{16} sin (4 x) + C

Solution steps

\int x sin (- 4 x) d x

Apply Integration By Parts

: - \int - 2 x^{2} cos (4 x) d x - \frac{1}{2} x^{2} sin (4 x)

= - \int - 2 x^{2} cos (4 x) d x - \frac{1}{2} x^{2} sin (4 x)

\int - 2 x^{2} cos (4 x) d x = - \frac{1}{16} (8 x^{2} sin (4 x) - sin (4 x) + 4 x cos (4 x))

= - (- \frac{1}{16} (8 x^{2} sin (4 x) - sin (4 x) + 4 x cos (4 x))) - \frac{1}{2} x^{2} sin (4 x)

Simplify

- (- \frac{1}{16} (8 x^{2} sin (4 x) - sin (4 x) + 4 x cos (4 x))) - \frac{1}{2} x^{2} sin (4 x) : \frac{1}{4} x cos (4 x) - \frac{1}{16} sin (4 x)

= \frac{1}{4} x cos (4 x) - \frac{1}{16} sin (4 x)

Add a constant to the solution

= \frac{1}{4} x cos (4 x) - \frac{1}{16} sin (4 x) + C

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What is the integral of xsin(-4x) ?
The integral of xsin(-4x) is 1/4 xcos(4x)-1/16 sin(4x)+C