Evaluate the Integral

Problem

(∫_1^4)((e√(,x))/√(,x)*d(x))

Solution

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

2. Calculate the differential d(u) by differentiating u with respect to x

d(u)/d(x)=1/2*x(−1/2)

d(u)=1/(2√(,x))*d(x)

2*d(u)=1/√(,x)*d(x)

3. Determine the new limits of integration for u
   When x=1 u=√(,1)=1
   When x=4 u=√(,4)=2

4. Substitute the variables and limits into the integral.

(∫_1^2)(2*eu*d(u))

5. Evaluate the definite integral using the fundamental theorem of calculus.

2*(∫_1^2)(eu*d(u))=2*[eu]21

2*(e2−e1)

6. Simplify the final expression.

2*e2−2*e

Final Answer

(∫_1^4)((e√(,x))/√(,x)*d(x))=2*e2−2*e

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