Solution
= Solution
Since the odd <sine> part of the complex exponential integrates to zero,
$$
\frac1{2\pi}\int_{-n}^ne^{ikx}\,dk
=\frac1\pi\int_0^n\cos(kx)\,dk.
$$
Part (e)(ii) therefore yields
$$
\boxed{\lim_{n\to\infty}\frac1{2\pi}\int_{-n}^ne^{ikx}\,dk=\delta(x)}
$$
in the sense of distributions. This is the <Fourier representation of the Dirac delta function>.