Evaluate the Integral

Problem

(∫_^)((e√(,x))/√(,x)*d(x))

Solution

1. Identify the substitution by choosing a variable u that simplifies the exponent. Let u=√(,x)

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

d(u)/d(x)=1/(2√(,x))

3. Rearrange the differential equation to solve for the terms present in the integral.

2*d(u)=1/√(,x)*d(x)

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

(∫_^)(eu⋅2*d(u))

5. Factor out the constant from the integral.

2*(∫_^)(eu*d(u))

6. Integrate the exponential function with respect to u

2*eu+C

7. Back-substitute the original variable x by replacing u with √(,x)

2*e√(,x)+C

Final Answer

(∫_^)((e√(,x))/√(,x)*d(x))=2*e√(,x)+C

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