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

Problem

(∫_^)(cos(2*x)*d(x))

Solution

1. Identify the integral as a basic trigonometric integral requiring a usubstitution or the reverse chain rule.

2. Substitute u=2*x to simplify the argument of the cosine function.

3. Differentiate u to find d(u)=2*d(x) which implies d(x)=1/2*d(u)

4. Rewrite the integral in terms of u

(∫_^)(cos(u)⋅1/2*d(u))

5. Integrate the function using the rule (∫_^)(cos(u)*d(u))=sin(u)+C

1/2*sin(u)+C

6. Back-substitute u=2*x to return to the original variable.

Final Answer

(∫_^)(cos(2*x)*d(x))=1/2*sin(2*x)+C

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