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

Problem

(∫_^)(csc2(x)*d(x))

Solution

1. Identify the standard trigonometric integral for the square of the cosecant function.

2. Recall the basic differentiation rule for the cotangent function, which states that d(cot(x))/d(x)=−csc2(x)

3. Apply the reverse power rule or standard integral table entry, noting that since the derivative of cot(x) is −csc2(x) the integral of csc2(x) must be −cot(x)

4. Add the constant of integration C to complete the indefinite integral.

Final Answer

(∫_^)(csc2(x)*d(x))=−cot(x)+C

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