What is the result of ∫ (1 + cos(2x)) dx?

To determine the integral of the function ( \int (1 + \cos(2x)) , dx ), we can break the integrand into two parts and evaluate each separately.

First, we handle the constant term ( 1 ). The integral of ( 1 ) with respect to ( x ) is simply ( x ):

[

\int 1 , dx = x.

]

Next, we need to integrate ( \cos(2x) ). To do this, we can use the substitution method. The integral of ( \cos(kx) ) is given by ( \frac{1}{k} \sin(kx) ). Here, ( k ) is ( 2 ):

[

\int \cos(2x) , dx = \frac{1}{2} \sin(2x).

]

Now, combining both results, we find:

[

\int (1 + \cos(2x)) , dx = \int 1 , dx + \int \cos(2x) , dx = x + \frac{1}{2} \sin(2x).

]

Finally, we add the constant