As is known, the classical theory of $C_{0}$-semigroups of linear operators is an important tool for studying many questions of the theory of differential equations in Banach space, in particular the Cauchy problem $C_{0}[\tau]$ of finding a solution $u(t), t \in [0, \tau]$ of the equation $u'(t) = Au(t)$ satisfying the condition $u(0) = x \in X$, where $A$ is a closed linear operator in the Banach space $X$. It turns out that one of the most fruitful methods for studying the $(n+1)$-times $(n \in \mathds{N})$ integrated Cauchy problem $C_{n+1}[\tau]: v'(t) = Av(t) + \frac{t^{n}}{n!}x, v(0) = 0$ is the study of the so-called $(n+1)$-times integrated semigroups introduced by Arendt, the theory of which was later developed by Kellerman and Heber, Tanaka and Miyadera, de Laubenfels, and others.

In this article, the main attention is focused on the case when $A$ is a normal operator in a Hilbert space. Based on the properties of the function ${\Phi}_n(\lambda, t) = \frac{1}{{\lambda}^{n + 1}}\left(e^{\lambda t} -
\sum\limits_{k = 0}^{n} \frac{(t\lambda)^k}{k!}\right), \lambda \in \mathbb{C}$, which is connected in a certain way with the corresponding $(n+1)$-times integrated semigroup, and the operational calculus for normal operators, using the specified function, all solutions of the problem $C_{n+1}[\tau]$ are described and necessary and sufficient conditions for its correct formulation are found. Moreover, a criterion for the correctness of this problem in terms of the localization of the spectrum of the operator $A$ is established.