Evaluate the Integral

Problem

(∫_^)(e^√(,x)*d(x))

Solution

1. Substitute a new variable to simplify the exponent by letting u=√(,x)

2. Differentiate the substitution to find the relationship between d(x) and d(u) where x=u^2 so d(x)=2*u*d(u)

3. Rewrite the integral in terms of u to get (∫_^)(2*u*e^u*d(u))

4. Apply integration by parts using the formula (∫_^)(ƒ*g^′=ƒ*g−(∫_^)(ƒ^′*g)) where ƒ=2*u and g^′=e^u

5. Calculate the parts: ƒ^′=2 and g=e^u leading to 2*u*e^u−(∫_^)(2*e^u*d(u))

6. Integrate the remaining term to get 2*u*e^u−2*e^u+C

7. Factor out the common terms to get 2*e^u*(u−1)+C

8. Back-substitute u=√(,x) to return to the original variable.

Final Answer

(∫_^)(e^√(,x)*d(x))=2*e^√(,x)*(√(,x)−1)+C

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