Evaluate the Integral

Problem

(∫_^)((e(1/x))/(x2)*d(x))

Solution

1. Identify the structure of the integrand and choose a substitution that simplifies the exponent.

2. Substitute u=1/x which can be written as u=x(−1)

3. Differentiate u with respect to x to find d(u)=−x(−2)*d(x) which simplifies to d(u)=−1/(x2)*d(x)

4. Rearrange the differential to solve for the terms in the integral: −d(u)=1/(x2)*d(x)

5. Rewrite the integral in terms of u

(∫_^)(−eu*d(u))

6. Integrate the expression with respect to u

−eu+C

7. Back-substitute the original expression for u to get the final result in terms of x

Final Answer

(∫_^)((e(1/x))/(x2)*d(x))=−e(1/x)+C

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