Evaluate the Integral integral of sec(4x)^2 with respect to x

Problem

(∫_^)(sec2(4*x)*d(x))

Solution

1. Identify the standard integral form for the square of the secant function, which is (∫_^)(sec2(u)*d(u))=tan(u)+C

2. Apply the substitution method by letting u=4*x

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

4. Substitute the values of u and d(x) into the integral to get (∫_^)(sec2(u)⋅1/4*d(u))

5. Factor out the constant 1/4 from the integral to get 1/4*(∫_^)(sec2(u)*d(u))

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

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

Final Answer

(∫_^)(sec2(4*x)*d(x))=1/4*tan(4*x)+C

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