In computer algebra (and in the Calculus classroom), it is important to know in advance if the integral $F(x) = \int_O^xf(t)dt$ of a given function $f$ can be expressed in terms of `known' functions, or if it defines a `new transcendent'. We here study the case when $f$ is a solution of a linear differential equation $Ly = O$ with coefficients in $C(z)$, all of whose solutions are viewed as `known' functions. By Picard-Vessiot theory (and an argument borrowed from Kummer theory), the problem translates into the study of the unipotent radical of differential Galois groups. We shall give various answers to the latter question, and illustrate them with applications to polylogarithms, to functional Mordell-Weil groups, and to self-dual operators.

 
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