  We show that the set of Lebesgue integrable functions in $[0,1]$ which are
nowhere essentially bounded is spaceable, improving a result from [F. J.
Garc\'{i}a-Pacheco, M. Mart\'{i}n, and J. B. Seoane-Sep\'ulveda.
\textit{Lineability, spaceability, and algebrability of certain subsets of
function spaces,} Taiwanese J. Math., \textbf{13} (2009), no. 4, 1257--1269],
and that it is strongly $\mathfrak{c}$-algebrable. We prove strong
$\mathfrak{c}$-algebrability and non-separable spaceability of the set of
functions of bounded variation which have a dense set of jump discontinuities.
Applications to sets of Lebesgue-nowhere-Riemann integrable and
Riemann-nowhere-Newton integrable functions are presented as corollaries. In
addition we prove that the set of Kurzweil integrable functions which are not
Lebesgue integrable is spaceable (in the Alexievicz norm) but not 1-algebrable.
We also show that there exists an infinite dimensional vector space $S$ of
differentiable functions such that each element of the $C([0,1])$-closure of
$S$ is a primitive to a Kurzweil integrable function, in connection to a
classic spaceability result from [V. I. Gurariy, \textit{Subspaces and bases in
spaces of continuous functions (Russian),} Dokl. Akad. Nauk SSSR, \textbf{167}
(1966), 971-973].
