Evaluate the Integral integral of cos(x)^2 with respect to x

Problem

(∫_^)(cos2(x)*d(x))

Solution

1. Identify the trigonometric identity for cos2(x) to reduce the power, as it cannot be integrated directly using basic rules.

2. Apply the identity cos2(x)=(1+cos(2*x))/2 to rewrite the integrand.

(∫_^)(cos2(x)*d(x))=(∫_^)((1+cos(2*x))/2*d(x))

3. Factor out the constant 1/2 from the integral.

1/2*(∫_^)((1+cos(2*x))*d(x))

4. Integrate the terms individually. The integral of 1 is x and the integral of cos(2*x) is sin(2*x)/2

1/2*(x+sin(2*x)/2)+C

5. Distribute the 1/2 to simplify the final expression.

x/2+sin(2*x)/4+C

Final Answer

(∫_^)(cos2(x)*d(x))=x/2+sin(2*x)/4+C

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