Integrate (sin(x))^2/x^2 using special functions if necessary
To integrate the function (\frac{\sin^2(x)}{x^2}), we can use a known result involving the sinc function. The integral can be expressed in terms of the sinc function, which is defined as:
[ \text{sinc}(x) = \frac{\sin(x)}{x} ]
Thus, we can rewrite the integral as:
[ \int \frac{\sin^2(x)}{x^2} , dx = \int \text{sinc}^2(x) , dx ]
The integral of (\text{sinc}^2(x)) can be evaluated using the following result:
[ \int \text{sinc}^2(x) , dx = \frac{x}{2} + \frac{1}{4} \sin(2x) + C ]
where (C) is the constant of integration.
Therefore, the integral of (\frac{\sin^2(x)}{x^2}) is:
[ \int \frac{\sin^2(x)}{x^2} , dx = \frac{x}{2} + \frac{1}{4} \sin(2x) + C ]
This result can be used for definite integrals or further analysis as needed.