Evaluate the Integral integral of cos(3x) with respect to x

Problem

(∫_^)(cos(3*x)*d(x))

Solution

1. Identify the integral form as a basic trigonometric integral requiring a usubstitution for the linear argument 3*x

2. Substitute u=3*x which implies that d(u)=3*d(x) or d(x)=1/3*d(u)

3. Rewrite the integral in terms of u

(∫_^)(cos(u)⋅1/3*d(u))

4. Factor out the constant 1/3 from the integral:

1/3*(∫_^)(cos(u)*d(u))

5. Integrate the cosine function, knowing that (∫_^)(cos(u)*d(u))=sin(u)+C

1/3*sin(u)+C

6. Back-substitute 3*x for u to return to the original variable:

1/3*sin(3*x)+C

Final Answer

(∫_^)(cos(3*x)*d(x))=1/3*sin(3*x)+C

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