What is the integral of (2+x)cos(2x) ?

The integral of (2+x)cos(2x) is sin(2x)+1/4 (2xsin(2x)+cos(2x))+C

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integral of (2+x)cos(2x)

\int (2 + x) cos (2 x) d x

sin (2 x) + \frac{1}{4} (2 x sin (2 x) + cos (2 x)) + C

Solution steps

\int (2 + x) cos (2 x) d x

Expand

(2 + x) cos (2 x) : 2 cos (2 x) + x cos (2 x)

= \int 2 cos (2 x) + x cos (2 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 2 cos (2 x) d x + \int x cos (2 x) d x

\int 2 cos (2 x) d x = sin (2 x)

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

= sin (2 x) + \frac{1}{4} (2 x sin (2 x) + cos (2 x))

Add a constant to the solution

= sin (2 x) + \frac{1}{4} (2 x sin (2 x) + cos (2 x)) + C

What is the integral of (2+x)cos(2x) ?
The integral of (2+x)cos(2x) is sin(2x)+1/4 (2xsin(2x)+cos(2x))+C