Choose $\Phi(t)=\sqrt{t^{2}+1}$ and observe that $\int_{M}e^{F}\sqrt{F^{2}+1}dvol_{g}$ is controlled in terms of an upper bound of $\int_{M}e^{F}Fdvol_{g}$. Then the result follows from Theorem 5.2 and Corollary 5.2. ∎