Integral of cos²(x)

We aim to find the integral of cos²(x). This is a common integral in trigonometry, and it’s often solved using trigonometric identities to simplify the expression.

Step 1: Use Trigonometric Identity

First, we use the trigonometric identity cos²(x) = (1 + cos(2x)) / 2 to rewrite cos²(x) in a more manageable form. This identity is derived from the double angle formulas for cosine.

cos²(x) = (1 + cos(2x)) / 2

Step 2: Rewrite the Integral Using the Identity

After applying the identity, the integral becomes:

∫ cos²(x) dx = ∫ (1 + cos(2x)) / 2 dx

Step 2.5: Factor Out the Constant 1/2

Before we proceed to integrate, we can factor out the constant 1/2 from the integral. In calculus, constants can be pulled out of integrals, which often simplifies the integration process.

= 1/2 ∫ (1 + cos(2x)) dx

Step 3: Integrate Term by Term

Now we integrate each term in the expression separately. The integral of 1 with respect to x is x. For the term cos(2x), the integral is sin(2x) / 4.

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

Here, C is the constant of integration, which can be any real number.

Final Result

Putting it all together, the integral of cos²(x) becomes 1/2 x + 1/4 sin(2x) + C. This is our final result, obtained by using a trigonometric identity to simplify the original integral, factoring out the constant 1/2, and then integrating each term separately.