Getting contradictory answers for the integrral of sin(x)/x?

Question

I wanted to find out the integral of sin(x)/x from negative infinity to infinity so I asked my physics/calculus teacher, who told me that it was equal to pi or pi/2, I can't remember which. Neither of us were able to prove it so I decided to use the fact that rect(f) is the fourier transform of sinc(t) but upon evaluating this at f = 0 I wound up with the answer that the integral is the square root of pi/2 ! Help please?

Answer 1

See the source cited below.

The value of the integral from -∞ to ∞ of sin(x)/x is π.

The discrepancy you ran into with using the Fourier transform may be due to the somewhat arbitrary way the factor of 2π is split up between the direct and inverse transforms. Also, as you will see from the Mathworld source, the Fourier transform of the rectangle function is sinc(πk).

The Mathworld source shows how to evaluate the integral using contour integration.