Evaluate the Integral

Problem

(∫_^)(sec2(1/x)/(x2)*d(x))

Solution

1. Identify the structure of the integrand and choose a substitution for the inner function u=1/x

2. Calculate the differential d(u) by differentiating u=x(−1) with respect to x

d(u)=−x(−2)*d(x)

d(u)=−1/(x2)*d(x)

3. Rearrange the differential to match the terms in the integral.

−d(u)=1/(x2)*d(x)

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

(∫_^)(−sec2(u)*d(u))

5. Integrate with respect to u using the standard integral (∫_^)(sec2(u)*d(u))=tan(u)

−tan(u)+C

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

−tan(1/x)+C

Final Answer

(∫_^)(sec2(1/x)/(x2)*d(x))=−tan(1/x)+C

Want more problems? Check here!