Evaluate the Integral

Problem

(∫_^)(sin(1/x)/(x2)*d(x))

Solution

1. Identify the substitution by looking for a function and its derivative within the integrand.

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

3. Differentiate u with respect to x to find d(u)

d(u)/d(x)=−x(−2)

d(u)/d(x)=−1/(x2)

4. Rearrange the differential to solve for the terms present in the integral.

−d(u)=1/(x2)*d(x)

5. Substitute u and −d(u) into the original integral.

(∫_^)(−sin(u)*d(u))

6. Integrate the trigonometric function with respect to u

−(∫_^)(sin(u)*d(u))=−(−cos(u))+C

cos(u)+C

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

cos(1/x)+C

Final Answer

(∫_^)(sin(1/x)/(x2)*d(x))=cos(1/x)+C

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