Evaluate the Integral

Problem

(∫_1^2)((e(1/x))/(x2)*d(x))

Solution

1. Identify a suitable substitution to simplify the integrand. Let u=1/x which can be written as u=x(−1)

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

d(u)/d(x)=−x(−2)

d(u)=−1/(x2)*d(x)

−d(u)=1/(x2)*d(x)

3. Determine the new limits of integration for u
   When x=1 u=1/1=1
   When x=2 u=1/2

4. Substitute the variables and limits into the integral.

(∫_1^1/2)(−eu*d(u))

5. Apply the property of integrals to reverse the limits and remove the negative sign.

(∫_1/2^1)(eu*d(u))

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

[eu]1(1/2)

e1−e(1/2)

e−√(,e)

Final Answer

(∫_1^2)((e(1/x))/(x2)*d(x))=e−√(,e)

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