How to Integrate cos²(x): A Step-by-Step Guide

Understanding the Problem

Integrating trigonometric functions can often be tricky, but with the right approach, even complex integrals can be simplified. Today, we're tackling the integral of cos²(x).

Step 1: The Double-Angle Formula

The key to solving this integral lies in a trigonometric identity known as the double-angle formula for cosine:

cos(2x) = 2cos²(x) - 1

Rearranging this formula to isolate cos²(x), we get:

cos²(x) = (1 + cos(2x)) / 2

Step 2: Substitute and Split

Now, we substitute this expression into our integral:

∫ cos²(x) dx = ∫ (1 + cos(2x)) / 2 dx

To simplify the integration, we can split this into two separate integrals:

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

Step 3: Integrate Each Part

The first integral is straightforward:

(1/2) ∫ 1 dx = (1/2)x + C₁

For the second integral, we'll use a substitution. Let u = 2x, so du = 2dx:

(1/2) ∫ cos(2x) dx = (1/4) ∫ cos(u) du = (1/4) sin(u) + C₂

Substituting back u = 2x:

(1/4) sin(2x) + C₂

Step 4: Combine the Results

Now, we combine the results from both integrals:

∫ cos²(x) dx = (1/2)x + (1/4) sin(2x) + C

Final Answer

Therefore, the integral of cos²(x) is:

∫ cos²(x) dx = (1/2)x + (1/4) sin(2x) + C

Where C is the constant of integration.

By following these steps and utilizing the double-angle formula, we've successfully integrated cos²(x). Remember, practice is key when it comes to mastering integration techniques.