Evaluate the Integral

Problem

(∫_^)(ex√(,1+ex)*d(x))

Solution

1. Identify a suitable substitution to simplify the integrand, choosing u to be the expression inside the square root.

u=1+ex

2. 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)

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

(∫_^)(√(,u)*d(u))

4. Rewrite the square root as a fractional exponent to prepare for integration.

(∫_^)(u(1/2)*d(u))

5. Apply the power rule for integration, which states (∫_^)(un*d(u))=(u(n+1))/(n+1)+C

(u(3/2))/(3/2)+C

6. Simplify the coefficient by multiplying by the reciprocal.

2/3*u(3/2)+C

7. Back-substitute the original expression for u to get the final result in terms of x

2/3*(1+ex)(3/2)+C

Final Answer

(∫_^)(ex√(,1+ex)*d(x))=2/3*(1+ex)(3/2)+C

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