Evaluate the Integral

Problem

(∫_^)(sec(2*x)*tan(2*x)*d(x))

Solution

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

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

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

4. Substitute the variables into the integral to get (∫_^)(sec(u)*tan(u)⋅1/2*d(u))

5. Factor out the constant 1/2 from the integral.

6. Integrate the expression with respect to u to obtain 1/2*sec(u)+C

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

Final Answer

(∫_^)(sec(2*x)*tan(2*x)*d(x))=1/2*sec(2*x)+C

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