What is the integral of cos² 2x?

The integral of cos²(2x) can be solved using a trigonometric identity. First, we apply the identity for cos²(θ):

Using the Trigonometric Identity

The identity states that:

• cos²(θ) = (1 + cos(2θ)) / 2

For our case, we replace θ with 2x:

• cos²(2x) = (1 + cos(4x)) / 2

Setting Up the Integral

Now, we can set up the integral:

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

Breaking It Down

This can be simplified to:

• ∫ (1/2) dx + ∫ (1/2) cos(4x) dx

Calculating Each Part

Now, we integrate each term separately:

• ∫ (1/2) dx = (1/2)x
• ∫ (1/2) cos(4x) dx = (1/8) sin(4x)

Combining the Results

Putting it all together, we have:

∫ cos²(2x) dx = (1/2)x + (1/8) sin(4x) + C

Final Answer

Thus, the integral of cos²(2x) is:

(1/2)x + (1/8) sin(4x) + C