Evaluate the Integral

Problem

(∫_^)(cos(π/x)/(x2)*d(x))

Solution

1. Identify the substitution method as the most efficient approach because the derivative of the inner function π/x is a multiple of the remaining term 1/(x2)

2. Substitute u=π/x which can be rewritten as u=π*x(−1)

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

d(u)/d(x)=−π*x(−2)

d(u)=−π/(x2)*d(x)

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

−1/π*d(u)=1/(x2)*d(x)

5. Rewrite the integral in terms of u

(∫_^)(cos(u)⋅(−1/π)*d(u))

−1/π*(∫_^)(cos(u)*d(u))

6. Integrate the cosine function with respect to u

−1/π*sin(u)+C

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

−1/π*sin(π/x)+C

Final Answer

(∫_^)(cos(π/x)/(x2)*d(x))=−sin(π/x)/π+C

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