Consider a random variable U that has a uniform distribution on (0,1) and a random variable $X$ that has a standard normal distribution. Assume that $U$ and $X$ are independent. Determine an expression for the probability density function of the random variable $Z=U+X$ in terms of the cumulative distribution function of $X$.
This post solves the problem by expanding the convolution as $$f_Z(z) = \int_0^1 f_U(u) f_X(z-u) \mathrm{d}u.$$
I can see the bounds of the integral are 'less messy' when expanding as above; however, why is expanding the convolution as follows invalid? $$f_Z(z) = \int_{-\infty}^\infty f_X(x) f_U(z-x) \mathrm{d}x.$$ $$f_Z(z) = \int_{-\infty}^\infty f_X(x)\mathrm{d}x.$$
In general, can some convolution problems only be solved by expanding the functions one way as a result of the constraints of the input bounds?