Is there a fundamental theorem of calculus for improper integrals?
Solution 1:
$$F(x) := -\int_x^\infty f(t)\,dt = -\int_x^c f(t)\;dt + \int_c^\infty f(t)\;dt$$
The derivative of the first term of this sum with respect to $x$ is $f(x)$, and that of the second term is $0.$ You have some leeway as to what $c$ is; you could just choose $a.$