Evaluate the Integral

Problem

(∫_^)((eu)/((1−eu)2)*d(u))

Solution

1. Identify a suitable substitution to simplify the denominator. Let v=1−eu

2. Differentiate the substitution to find the relationship between d(v) and d(u)

d(v)/d(u)=−eu

d(v)=−eu*d(u)

−d(v)=eu*d(u)

3. Substitute the expressions for v and d(v) into the original integral.

(∫_^)(1/(v2)*(−d(v)))

4. Rewrite the integral using power notation for easier integration.

−(∫_^)(v(−2)*d(v))

5. Integrate using the power rule (∫_^)(vn*d(v))=(v(n+1))/(n+1)+C

−((v(−1))/(−1))+C

1/v+C

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

1/(1−eu)+C

Final Answer

(∫_^)((eu)/((1−eu)2)*d(u))=1/(1−eu)+C

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