Evaluate the Integral integral of cot(x) with respect to x

Problem

(∫_^)(cot(x)*d(x))

Solution

1. Rewrite the cotangent function using the fundamental trigonometric identity cot(x)=cos(x)/sin(x)

(∫_^)(cos(x)/sin(x)*d(x))

2. Identify a substitution where the numerator is the derivative of the denominator. Let u=sin(x)

u=sin(x)

3. Differentiate u with respect to x to find d(u)

d(u)/d(x)=cos(x)

d(u)=cos(x)*d(x)

4. Substitute u and d(u) into the integral.

(∫_^)(1/u*d(u))

5. Integrate using the natural logarithm rule (∫_^)(1/u*d(u))=ln(u)+C

ln(u)+C

6. Back-substitute the original expression for u to get the final result in terms of x

ln(sin(x))+C

Final Answer

(∫_^)(cot(x)*d(x))=ln(sin(x))+C

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