Evaluate the Integral

Problem

(∫_^)(cos(2*x)/(1+sin(2*x))*d(x))

Solution

1. Identify the structure of the integrand and choose a substitution u such that its derivative appears in the numerator.

2. Substitute u=1+sin(2*x)

3. Differentiate u with respect to x to find d(u)

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

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

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

5. Rewrite the integral in terms of u

(∫_^)(1/u⋅1/2*d(u))

6. Integrate using the natural logarithm rule (∫_^)(1/u*d(u))=ln(u)+C

1/2*ln(u)+C

7. Back-substitute the original expression for u

1/2*ln(1+sin(2*x))+C

Final Answer

(∫_^)(cos(2*x)/(1+sin(2*x))*d(x))=1/2*ln(1+sin(2*x))+C

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