Evaluate the Integral integral of cos(4x) with respect to x

Problem

(∫_^)(cos(4*x)*d(x))

Solution

1. Identify the integral as a basic trigonometric integral requiring a usubstitution because the argument of the cosine function is 4*x rather than just x

2. Substitute u=4*x to simplify the integrand.

3. Differentiate u with respect to x to find the relationship between d(u) and d(x)

d(u)/d(x)=4

⇒d(x)=1/4*d(u)

4. Rewrite the integral in terms of u by substituting 4*x with u and d(x) with 1/4*d(u)

(∫_^)(cos(u)⋅1/4*d(u))

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

1/4*(∫_^)(cos(u)*d(u))

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

1/4*sin(u)+C

7. Back-substitute the original expression 4*x for u to get the final result in terms of x

Final Answer

(∫_^)(cos(4*x)*d(x))=1/4*sin(4*x)+C

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