Calculate the integral ∫ cos^2(x) dx.

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Question: Calculate the integral ∫ cos^2(x) dx.

Options:

Correct Answer: (1/2)x + (1/4)sin(2x) + C

Solution:

Using the identity cos^2(x) = (1 + cos(2x))/2, we find that ∫ cos^2(x) dx = (1/2)x + (1/4)sin(2x) + C.

Calculate the integral ∫ cos^2(x) dx.

Calculate the integral ∫ cos^2(x) dx.

Step 1: Recall the integral we want to calculate: ∫ cos^2(x) dx.
Step 2: Use the trigonometric identity: cos^2(x) = (1 + cos(2x))/2.
Step 3: Substitute the identity into the integral: ∫ cos^2(x) dx = ∫ (1 + cos(2x))/2 dx.
Step 4: Split the integral into two parts: ∫ (1/2) dx + ∫ (1/2) cos(2x) dx.
Step 5: Calculate the first integral: ∫ (1/2) dx = (1/2)x.
Step 6: Calculate the second integral: ∫ (1/2) cos(2x) dx. The integral of cos(2x) is (1/2)sin(2x), so this becomes (1/4)sin(2x).
Step 7: Combine the results from Step 5 and Step 6: (1/2)x + (1/4)sin(2x).
Step 8: Don't forget to add the constant of integration, C: ∫ cos^2(x) dx = (1/2)x + (1/4)sin(2x) + C.

Trigonometric Identities – Understanding and applying the identity for cos^2(x) to simplify the integral.
Integration Techniques – Using substitution and integration of trigonometric functions to solve the integral.