Final answer:
The integral ∫[infinity]₋[infinity](sin(200πt)/t) ²dt can be evaluated using the Fourier transform, converting the time-domain signal into the frequency domain and finding the area under the power spectral density function.
Step-by-step explanation:
We need to evaluate the integral ∫[infinity]₋[infinity](sin(200πt)/t) ²dt. To simplify the problem, we can use a property of the Fourier transform, which allows us to convert a time-domain signal into its frequency-domain representation. In frequency domain, the power spectrum of a signal is the square of its Fourier transform magnitude. For a time-domain signal f(t), the power spectrum P(ω) is given by |F(ω)|², where F(ω) is the Fourier transform of f(t).
In our case, we have a function (sin(200πt)/t), which closely resembles the sinc function, commonly encountered in signal processing. The Fourier transform of a sinc function is a rectangular function. Therefore, the integral given is essentially the energy of the signal, which can also be interpreted as the area under the power spectral density function in the frequency domain. This area can be found by integrating over the frequency band that the rectangular function spans, which is a much simpler integral to calculate.
So, instead of directly computing the complex integral in the time domain, we can transform it into the easier task of finding the area under a rectangle in the frequency domain, and in our case, this area is simply the product of the height and width of the rectangle.