Convolution of two PDF

Question

I am probably overlooking like, all the important details, but when trying to work out how to take the convolution of two pdfs I am going as follows:

1. according to https://en.wikipedia.org/wiki/Convolution there is an identity $\int_{\mathbb{R}^{d}}(f*g)(x)dx = (\int_{\mathbb{R}^{d}}f(x)dx)(\int_{\mathbb{R}^{d}}g(x)dx) $
2. Let p = f(x) and q = g(x), and P, Q be their CDF respectively
3. $\int_{\mathbb{R}^{d}}(p*q)dx = O = PQ $
4. $p*q = \frac{d}{dx}PQ = pQ + qP$

But this doesn't seem to work even in the most trivial case of a uniform distribution on 0, 1 with itself, so ignoring any domain restrictions

$p = 1, q = 1, P = x, Q = x$

$pQ + qP = x + x = 2x$

But it should be the same uniform distribution

Can anyone explain to me where I am going wrong and whether there is a better way to get the convolution of two pdfs without using the standard convolution integration definition