**Integration of cos^2x**

To find the integral of cos^2x, we can use various methods such as substitution or trigonometric identities. Let's explore two common approaches: using the power reduction formula and using the double angle formula.

**Using the Power Reduction Formula**

The power reduction formula states that:
cos^2x = (1 + cos2x)/2

We can rewrite the integral as:
∫ cos^2x dx = ∫ (1 + cos2x)/2 dx

Now, let's split the integral into two parts:
∫ (1 + cos2x)/2 dx = ∫ (1/2) dx + ∫ (cos2x/2) dx

Integrating each part separately:
∫ (1/2) dx = (1/2) x + C1, where C1 is the constant of integration.

To integrate the second part, we can use the substitution method. Let's substitute u = 2x:
∫ (cos2x/2) dx = (1/2) ∫ cosu du

Now, integrating cosu:
(1/2) ∫ cosu du = (1/2) sinu + C2, where C2 is another constant of integration.

Substituting back the value of u:
(1/2) sinu + C2 = (1/2) sin(2x) + C2

Therefore, the final integral of cos^2x is:
∫ cos^2x dx = (1/2) x + (1/4) sin(2x) + C

**Using the Double Angle Formula**

Another approach to integrate cos^2x is by using the double angle formula:
cos^2x = (1 + cos2x)/2

We can rewrite the integral as:
∫ cos^2x dx = ∫ (1 + cos2x)/2 dx

Again, let's split the integral into two parts:
∫ (1 + cos2x)/2 dx = ∫ (1/2) dx + ∫ (cos2x/2) dx

Integrating each part separately:
∫ (1/2) dx = (1/2) x + C1, where C1 is the constant of integration.

To integrate the second part, we can use the double angle formula: cos2x = 2cos^2x - 1.
∫ (cos2x/2) dx = (1/2) ∫ (2cos^2x - 1)/2 dx

Simplifying the integral:
(1/2) ∫ (2cos^2x - 1)/2 dx = (1/2) ∫ (cos^2x - 1/2) dx

Integrating each part separately:
(1/2) ∫ (cos^2x - 1/2) dx = (1/2) ∫ cos^2x dx - (1/2) ∫ dx

The integral of dx is simply x, and the integral of cos^2x, as we derived earlier, is (1/2) x + C1.

Thus, the final integral of cos^2x is:
∫ cos^2x dx = (1/2) x - (1/4)