The integral of cos^2x can be evaluated using trigonometric identities.
Recall the identity: cos^2x = (1/2) + (1/2)cos(2x)
Therefore, we can write:
- ∫cos^2x dx = ∫[(1/2) + (1/2)cos(2x)] dx
Using the linearity of integration, we can split this into two integrals:
- ∫(1/2)dx + ∫(1/2)cos(2x) dx
Evaluating these integrals, we get:
- (1/2)x + (1/4)sin(2x) + C
where C is the constant of integration.
Therefore, the integral of cos^2x is equal to (1/2)x + (1/4)sin(2x) + C
Here are the steps to find the integral of cos^2x:
1. Use the trigonometric identity cos^2x = (1/2) + (1/2)cos(2x) to write the integrand in a simpler form.
2. Split the integral into two parts using the linearity of integration:
- ∫cos^2x dx = ∫[(1/2) + (1/2)cos(2x)] dx = (1/2)∫dx + (1/2)∫cos(2x) dx
3. Evaluate each integral:
- ∫dx = x + C1 (where C1 is the constant of integration)
- ∫cos(2x) dx = (1/2)sin(2x) + C2 (where C2 is the constant of integration)
4. Combine the results of the two integrals:
∫cos^2x dx = (1/2)∫dx + (1/2)∫cos(2x) dx = (1/2)(x + C1) + (1/2)(1/2)sin(2x) + C2 = (1/2)x + (1/4)sin(2x) + C (where C = (1/2)C1 + C2 is the constant of integration)
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