Evaluate the Integral

Problem

(∫_^)(cot(x)*ln(sin(x))*d(x))

Solution

1. Identify the substitution method as the most efficient approach because the derivative of ln(sin(x)) is cot(x)

2. Set a new variable u=ln(sin(x))

3. Differentiate u with respect to x using the chain rule: d(u)/d(x)=1/sin(x)⋅cos(x)=cot(x)

4. Substitute u and d(u)=cot(x)*d(x) into the original integral to get (∫_^)(u*d(u))

5. Integrate with respect to u using the power rule: (u2)/2+C

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

Final Answer

(∫_^)(cot(x)*ln(sin(x))*d(x))=((ln(sin(x)))2)/2+C

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