The integration of the sinc function, specifically the integral \(\int_a^b \left[\frac {\sin x}{x}\right]^2 dx\), can effectively be approached using power series methods. The Taylor series expansion for \(\sin^2 x\) is given by \(-\sum^\infty_{k=1} \frac{(-1)^{k} 2^{2k-1}x^{2k}}{(2k)!}\). By substituting this series into the sinc function and applying term-wise integration along with the fundamental theorem of calculus, one can derive a general form for the integral as a power series.
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