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

Problem

(∫_^)(csc(2*x)*d(x))

Solution

1. Identify the standard integral form for the cosecant function, which is (∫_^)(csc(u)*d(u))=−ln(csc(u)+cot(u))+C or (∫_^)(csc(u)*d(u))=ln(csc(u)−cot(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 (∫_^)(csc(u)1/2*d(u))

5. Factor out the constant 1/2 to get 1/2*(∫_^)(csc(u)*d(u))

6. Integrate with respect to u using the formula 1/2*ln(csc(u)−cot(u))+C

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

Final Answer

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

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