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

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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

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