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

Problem

(∫_^)(sec(4*x)*d(x))

Solution

1. Identify the standard integral form for the secant function, which is (∫_^)(sec(u)*d(u))=ln(sec(u)+tan(u))+C

2. Apply a substitution by letting u=4*x

3. Differentiate the substitution to find d(u)=4*d(x) which implies d(x)=1/4*d(u)

4. Substitute the variables into the integral to get (∫_^)(sec(u)⋅1/4*d(u))

5. Factor out the constant 1/4 to get 1/4*(∫_^)(sec(u)*d(u))

6. Integrate with respect to u to obtain 1/4*ln(sec(u)+tan(u))+C

7. Back-substitute u=4*x to express the final result in terms of x

Final Answer

(∫_^)(sec(4*x)*d(x))=1/4*ln(sec(4*x)+tan(4*x))+C

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