Evaluate the Integral

Problem

(∫_^)(1−csc(x)*cot(x)*d(x))

Solution

1. Split the integral using the sum/difference rule for integration.

(∫_^)(1−csc(x)*cot(x)*d(x))=(∫_^)(1*d(x))−(∫_^)(csc(x)*cot(x)*d(x))

2. Integrate the constant term with respect to x

(∫_^)(1*d(x))=x

3. Apply the trigonometric integral rule for the product of cosecant and cotangent.

(∫_^)(csc(x)*cot(x)*d(x))=−csc(x)

4. Combine the results and include the constant of integration C

x−(−csc(x))+C

5. Simplify the signs.

x+csc(x)+C

Final Answer

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

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