Compute the integral ∫ (1 + cos^2(x)) dx.

To compute the integral ∫ (1 + cos^2(x)) dx, we start by breaking the integral into two parts:

∫ (1 + cos^2(x)) dx = ∫ 1 dx + ∫ cos^2(x) dx

The first part, ∫ 1 dx, is straightforward and yields x.

For the second part, ∫ cos^2(x) dx, we can use the trigonometric identity:

cos^2(x) = (1 + cos(2x))/2

Now substituting this identity into the integral, we have:

∫ cos^2(x) dx = ∫ (1 + cos(2x))/2 dx

This separates further into:

= (1/2) ∫ 1 dx + (1/2) ∫ cos(2x) dx

The first part, (1/2) ∫ 1 dx, results in (1/2)x. The second part requires evaluating ∫ cos(2x) dx, which is (1/2)sin(2x) (since the derivative of sin(2x) is 2cos(2x), we need to account for that factor of 2