∫ cos²x dx

To solve the integral of cos²x, we can use a trigonometric identity to simplify the expression. The identity states that:

Trigonometric Identity

cos²x can be rewritten using the identity:

• cos²x = (1 + cos(2x)) / 2

Setting Up the Integral

Now, we can substitute this identity into the integral:

∫ cos²x dx = ∫ (1 + cos(2x)) / 2 dx

Breaking It Down

This can be separated into two simpler integrals:

∫ cos²x dx = 1/2 ∫ 1 dx + 1/2 ∫ cos(2x) dx

Calculating Each Integral

• The integral of 1 is simply x.
• The integral of cos(2x) is (1/2)sin(2x).

Putting It All Together

Now, we can combine the results:

∫ cos²x dx = 1/2 * x + 1/4 * sin(2x) + C

Final Result

Thus, the integral of cos²x is:

∫ cos²x dx = (1/2)x + (1/4)sin(2x) + C