Evaluate the Integral integral of csc(3x)^2 with respect to x

Problem

(∫_^)(csc2(3*x)*d(x))

Solution

1. Identify the standard integral form for the cosecant squared function, which is (∫_^)(csc2(u)*d(u))=−cot(u)+C

2. Apply the substitution method by letting u=3*x

3. Differentiate u with respect to x to find d(u)=3*d(x) which implies d(x)=1/3*d(u)

4. Substitute the variables into the integral to get (∫_^)(csc2(u)⋅1/3*d(u))

5. Factor out the constant 1/3 from the integral.

6. Integrate the expression to obtain −1/3*cot(u)+C

7. Substitute back the original variable u=3*x into the result.

Final Answer

(∫_^)(csc2(3*x)*d(x))=−1/3*cot(3*x)+C

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