Evaluate the Integral integral of cot(2x) with respect to x

Problem

(∫_^)(cot(2*x)*d(x))

Solution

1. Rewrite the cotangent function in terms of sine and cosine using the identity cot(θ)=cos(θ)/sin(θ)

(∫_^)(cos(2*x)/sin(2*x)*d(x))

2. Identify a substitution to simplify the integral. Let u=sin(2*x)

u=sin(2*x)

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

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

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

4. Adjust the differential to match the expression in the integral.

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

5. Substitute u and d(u) into the integral.

(∫_^)(1/u⋅1/2*d(u))

1/2*(∫_^)(1/u*d(u))

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

1/2*ln(u)+C

7. Back-substitute the original expression for u

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

Final Answer

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

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