Integrate the function (3-5x)*cos4x.

One of the techniques for evaluating integrals is
integration by parts.

We'll express the formula of
integration by parts using differentials:

Int u dv = u*v -
Int v du

We'll put u = 3 -
5x

We'll differentiate both
sides:

du = -5dx

We'll put dv
= cos(4x) dx.

We'll integrate both
sides:

Int dv = Int cos(4x)
dx

v = (sin 4x)/4 

We'll
substitute in the formula of integral:

Int (3 - 5x) cos(4x)
dx = (3 - 5x)*(sin 4x)/4  + 5 Int (sin 4x)dx/4 

Int (3 -
5x) cos(4x) dx = (3 - 5x)*(sin 4x)/4  + (5/4)Int (sin
4x)dx

Int (3 - 5x) cos(4x) dx = (3 - 5x)*(sin 4x)/4  +
(5/4)[(1/4)(- cos 4x)] + C

Int (3 - 5x)
cos(4x) dx = (3 - 5x)*(sin 4x)/4  - (5/16)[(cos 4x)] +
C