Evaluate the Integral

Problem

(∫_^)(sin(2*x)*cos(2*x)*d(x))

Solution

1. Identify the structure of the integrand as a power of a function multiplied by its derivative, suggesting the use of usubstitution.

2. Choose a substitution variable u=sin(2*x)

3. Differentiate u with respect to x to find d(u) applying the chain rule:

d(u)/d(x)=2*cos(2*x)

4. Rearrange the differential to solve for the terms present in the integral:

1/2*d(u)=cos(2*x)*d(x)

5. Substitute u and 1/2*d(u) into the original integral:

(∫_^)(u5⋅1/2*d(u))

6. Factor out the constant from the integral:

1/2*(∫_^)(u5*d(u))

7. Apply the power rule for integration, (∫_^)(un*d(u))=(u(n+1))/(n+1)+C

1/2⋅(u6)/6+C

8. Simplify the expression:

(u6)/12+C

9. Back-substitute the original expression for u

sin(2*x)/12+C

Final Answer

(∫_^)(sin(2*x)*cos(2*x)*d(x))=sin(2*x)/12+C

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