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

Problem

(∫_^)(2*cos2(x)*d(x))

Solution

1. Identify the trigonometric identity for cos2(x) to simplify the integrand.

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

3. Substitute the identity into the integral:

(∫_^)(2*((1+cos(2*x))/2)*d(x))

4. Simplify the integrand by canceling the constant factor 2:

(∫_^)(1+cos(2*x)*d(x))

5. Integrate term by term using the power rule and the rule for cosine:

(∫_^)(1*d(x))+(∫_^)(cos(2*x)*d(x))

6. Evaluate the integrals, remembering to divide by the coefficient of x for the trigonometric term and add the constant of integration C

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

Final Answer

(∫_^)(2*cos2(x)*d(x))=x+sin(2*x)/2+C

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