Question 2016.S2.29: The value of the integral 2 ∫-∞^∞ (sin 2πt/πt) dt is equal t......

We know that \frac{\sin 2 \pi t}{\pi t} is an even function.
Therefore,

\begin{aligned}& 2 \int_{-\infty}^{\infty} \frac{\sin (2 \pi t)}{\pi t} d t=2 \times 2 \int_0^{\infty} \frac{\sin (2 \pi t)}{\pi t} d t \\& \int^{\infty} \frac{\sin (2 \pi t)}{\pi t} d t=\frac{1}{2}\end{aligned}

Therefore,

\int^{\infty} \frac{\sin (2 \pi t)}{2 \pi t}=1