Evaluate the Integral

Problem

(∫_^)(sin(x)/(1+cos(x))*d(x))

Solution

1. Identify a suitable substitution by noticing that the derivative of the denominator is related to the numerator.

2. Substitute u=1+cos(x) which implies that d(u)=−sin(x)*d(x) or −d(u)=sin(x)*d(x)

3. Rewrite the integral in terms of u

(∫_^)((−1)/u*d(u))

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

−ln(u)+C

5. Back-substitute the original expression for u

−ln(1+cos(x))+C

Final Answer

(∫_^)(sin(x)/(1+cos(x))*d(x))=−ln(1+cos(x))+C

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