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We consider a family of tronqu\'{e}e solutions of the Painelv\'{e} IIequation \begin{equation*} q''(s)=2q(s)^3+sq(s)-(2\alpha+\frac12), \qquad\alpha > -\frac12, \end{equation*} which is characterized by the Stokesmultipliers $$s_1=-e^{-2\alpha \pi i },\quad s_2=\omega, \quad s_3=-e^{2 \alpha\pi i} $$ with $\omega$ being a free parameter. These solutions include thewell-known generalized Hastings-McLeod solution as a special case if$\omega=0$. We derive asymptotics of integrals of the tronqu\'{e}e solutionsand the associated Hamiltonians over the real axis for $\alpha > -1/2$ and$\omega \geq 0$, with the constant terms evaluated explicitly. Our resultsagree with those already known in the literature if the parameters $\alpha$ and$\omega$ are chosen to be special values. Some applications of our results inrandom matrix theory are also discussed.

On integrals of the tronqu\'{e}e solutions and the associated Hamiltonians for the Painlev\'{e} II equation