Let $f(x) = e^{x}$, which, expressed as a Maclaurin series, is equal to:
$$\sum_{i = 0}^{\infty}{\frac{x^{i}}{i!}}$$
Therefore, $f(-t^{2})$ gives:
$$\sum_{i = 0}^{\infty}{\frac{(-t^{2})^{i}}{i!}} = \sum_{i = 0}^{\infty}{\frac{(-1)^{i}t^{2i}}{i!}}$$
Therefore, $\int{f(-t^{2})\,dt}$ gives:
$$\sum_{i = 0}^{\infty}{\frac{(-1)^{i}}{i!}\int{t^{2i}\,dt}} = \sum_{i = 0}^{\infty}{\frac{(-1)^{i}t^{2i + 1}}{i!(2i + 1)}}$$
Is this correct?