Evaluate the Integral

Problem

(∫_^)(cos(1/x)/(x2)*d(x))

Solution

1. Identify the structure of the integrand and notice that the derivative of the inner function 1/x is present as −1/(x2)

2. Apply substitution by letting u=1/x

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

d(u)/d(x)=−1/(x2)

d(u)=−1/(x2)*d(x)

−d(u)=1/(x2)*d(x)

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

(∫_^)(cos(u)*(−d(u)))

−(∫_^)(cos(u)*d(u))

5. Integrate the function with respect to u

−sin(u)+C

6. Back-substitute the original expression u=1/x to find the final result in terms of x

Final Answer

(∫_^)(cos(1/x)/(x2)*d(x))=−sin(1/x)+C

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