What is the integral of (3-5x)sin(4x) ?

The integral of (3-5x)sin(4x) is -3/4 cos(4x)-5(-1/4 xcos(4x)+1/16 sin(4x))+C

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integral of (3-5x)sin(4x)

\int (3 - 5 x) sin (4 x) d x

- \frac{3}{4} cos (4 x) - 5 (- \frac{1}{4} x cos (4 x) + \frac{1}{16} sin (4 x)) + C

Solution steps

\int (3 - 5 x) sin (4 x) d x

Expand

(3 - 5 x) sin (4 x) : 3 sin (4 x) - 5 x sin (4 x)

= \int 3 sin (4 x) - 5 x sin (4 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 3 sin (4 x) d x - \int 5 x sin (4 x) d x

\int 3 sin (4 x) d x = - \frac{3}{4} cos (4 x)

\int 5 x sin (4 x) d x = 5 (- \frac{1}{4} x cos (4 x) + \frac{1}{16} sin (4 x))

= - \frac{3}{4} cos (4 x) - 5 (- \frac{1}{4} x cos (4 x) + \frac{1}{16} sin (4 x))

Add a constant to the solution

= - \frac{3}{4} cos (4 x) - 5 (- \frac{1}{4} x cos (4 x) + \frac{1}{16} sin (4 x)) + C

What is the integral of (3-5x)sin(4x) ?
The integral of (3-5x)sin(4x) is -3/4 cos(4x)-5(-1/4 xcos(4x)+1/16 sin(4x))+C