Evaluate the Integral

Problem

(∫_^)((e√(,y))/√(,y)*d(y))

Solution

1. Identify the substitution to simplify the exponent. Let u=√(,y) which can also be written as u=y(1/2)

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

d(u)/d(y)=1/2*y(−1/2)

d(u)/d(y)=1/(2√(,y))

3. Solve for d(y) or rearrange the differential to match the integrand.

2*d(u)=1/√(,y)*d(y)

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

(∫_^)(eu⋅2*d(u))

2*(∫_^)(eu*d(u))

5. Integrate the function with respect to u

2*eu+C

6. Back-substitute the original variable y by replacing u with √(,y)

2*e√(,y)+C

Final Answer

(∫_^)((e√(,y))/√(,y)*d(y))=2*e√(,y)+C

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