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

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

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integral of (x-4)cos(4x)

\int (x - 4) cos (4 x) d x

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

Solution steps

\int (x - 4) cos (4 x) d x

Expand

(x - 4) cos (4 x) : x cos (4 x) - 4 cos (4 x)

= \int x cos (4 x) - 4 cos (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 x cos (4 x) d x - \int 4 cos (4 x) d x

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

\int 4 cos (4 x) d x = sin (4 x)

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

Add a constant to the solution

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

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