Evaluate the Integral

Problem

(∫_^)((ex)/(1+ex)*d(x))

Solution

1. Identify the substitution method as the most efficient approach because the numerator ex is the derivative of the denominator's term ex

2. Substitute a new variable u for the denominator.

u=1+ex

3. Differentiate u with respect to x to find the relationship between d(u) and d(x)

d(u)/d(x)=ex

d(u)=ex*d(x)

4. Rewrite the integral in terms of u by substituting 1+ex with u and ex*d(x) with d(u)

(∫_^)(1/u*d(u))

5. Integrate the expression using the natural logarithm rule.

ln(u)+C

6. Back-substitute the original expression for u to return to the variable x Since 1+ex is always positive, the absolute value bars can be replaced with parentheses.

ln(1+ex)+C

Final Answer

(∫_^)((ex)/(1+ex)*d(x))=ln(1+ex)+C

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