Fourier transform of $ \log(x^{2}+a^{2}) $ I would like to evaluate the Fourier cosine transform of $\log(x^{2}+a^{2})$
or the integral
$$\int_{0}^{\infty}\cos(ux)\log(x^{2}+a^{2})\,dx$$
for any real $u,a$.
However, it seems that this integral may be divergent due to the behaviour of the logarithm so what should I do?

For an even function, the two-sided cosine transform coincides with the complex exponential transform, so the answer will be the same as here:
$$\int_{-\infty}^\infty \ln\left( x^2+a^2 \right) \cos u x \,dx =
-2 \pi \left( e^{-\left| a u \right|} \left| u \right|^{-1} +
 2 \gamma \delta(u) \right),$$
which can be verified by computing the inverse transform.
For a one-sided transform, first you would have to extend the definition of the transform itself to make it applicable to distributions, because the integral from zero to infinity of the delta function is undefined.
