Evaluate the Integral

Problem

(∫_^)((eu)/((6−eu)2)*d(u))

Solution

1. Identify a suitable substitution to simplify the denominator. Let v=6−eu

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

d(v)=−eu*d(u)

3. Rearrange the differential to solve for the numerator in the integral.

−d(v)=eu*d(u)

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

(∫_^)((−1)/(v2)*d(v))

5. Rewrite the integrand using a negative exponent to prepare for the power rule.

−(∫_^)(v(−2)*d(v))

6. Integrate using the power rule for integration.

−((v(−1))/(−1))+C

7. Simplify the resulting expression.

1/v+C

8. Back-substitute the original expression for v to return to the variable u

1/(6−eu)+C

Final Answer

(∫_^)((eu)/((6−eu)2)*d(u))=1/(6−eu)+C

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