Evaluate ∫ cos^2(x) dx.

To evaluate the integral ∫ cos²(x) dx, it's beneficial to use a trigonometric identity to simplify the expression. The identity that will help in this case is:

[

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

]

This transformation allows us to rewrite the integral:

[

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

]

Now we can separate this into two simpler integrals:

[

\int \cos^2(x) , dx = \frac{1}{2} \int (1 + \cos(2x)) , dx = \frac{1}{2} \left( \int 1 , dx + \int \cos(2x) , dx \right)

]

Calculating each part gives:

1. The integral of 1 with respect to x is simply x.

2. The integral of cos(2x) can be solved using the substitution method or knowing the antiderivative directly, which gives us (\frac{1}{2} \sin(2x)\