Find the Antiderivative cot(x)

Problem

(∫_^)(cot(x)*d(x))

Solution

1. Rewrite the cotangent function in terms of sine and cosine using the quotient identity.

(∫_^)(cot(x)*d(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)=cos(x)*d(x)

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

(∫_^)(1/u*d(u))

5. Integrate using the natural logarithm rule for the reciprocal function.

ln(u)+C

6. Back-substitute the original expression for u to get the final result.

ln(sin(x))+C

Final Answer

(∫_^)(cot(x)*d(x))=ln(sin(x))+C

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