Integration of 1/ x^2 sin^2(1/x) dx

Understanding the Integral:

To evaluate the given integral, we need to understand the properties of the function sin^2(1/x) and how it affects the integration process. The function sin^2(1/x) is periodic and oscillatory, which can make the integration challenging.

Integration by Substitution:

1. Let's start by making a substitution to simplify the integral.
2. Substitute u = 1/x, then du = -1/x^2 dx.
3. The integral becomes ∫ -sin^2(u) du.

Trigonometric Identity:

1. We can use the trigonometric identity sin^2(u) = 1/2 - 1/2 cos(2u).
2. Substituting this identity into the integral, we get ∫ -(1/2 - 1/2 cos(2u)) du.

Integration of the Terms:

1. Integrate the terms separately to get -u/2 + 1/4 sin(2u) + C.
2. Substitute back u = 1/x to get the final result in terms of x.
3. The final result of the integral is -1/(2x) + 1/4 sin(2/x) + C.

Conclusion:

In conclusion, by using the substitution method and trigonometric identities, we were able to evaluate the integral of 1/x^2 sin^2(1/x) dx. The final result is -1/(2x) + 1/4 sin(2/x) + C.