Find the Integral cos(x)^2

Problem

(∫_^)(cos2(x)*d(x))

Solution

1. Apply the power-reduction identity to rewrite the integrand using the formula cos2(x)=(1+cos(2*x))/2

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

2. Factor out the constant 1/2 from the integral to simplify the expression.

1/2*(∫_^)((1+cos(2*x))*d(x))

3. Distribute the integral across the sum of the two terms.

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

4. Evaluate the integrals using the basic rules (∫_^)(1*d(x))=x and (∫_^)(cos(a*x)*d(x))=sin(a*x)/a

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

5. Distribute the constant to reach the final simplified form.

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

Final Answer

(∫_^)(cos2(x)*d(x))=x/2+sin(2*x)/4+C

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