Evaluate the Integral integral of csc(5x) with respect to x

Problem

(∫_^)(csc(5*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

2. Apply a substitution by letting u=5*x

3. Calculate the differential d(u)=5*d(x) which implies d(x)=1/5*d(u)

4. Substitute the variables into the integral to get (∫_^)(csc(u)⋅1/5*d(u))

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

6. Integrate with respect to u to obtain 1/5*(−ln(csc(u)+cot(u)))+C

7. Substitute back u=5*x to express the final result in terms of x

Final Answer

(∫_^)(csc(5*x)*d(x))=−1/5*ln(csc(5*x)+cot(5*x))+C

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