A Naturally Limited Integrand

Just use L-Hospital 2 times Using the Newton-Leibnitz rule in the first step of differentiation

The Mclaurin expansion of $e^x$ about 0 is $\displaystyle 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...$

Now setting $x=-t^2$ and integrating: $\int_0^x1-t^2+\frac{t^4}{2!}-\frac{t^6}{3!}+\frac{t^8}{4!}-...dt=x-\frac{x^3}{3}+\frac{x^5}{5\times2!}-\frac{x^7}{7\times3!}+...$

The first terms cancel as well as after the third, and all that is left is $\displaystyle\frac{x^5}{x^5*5*2!}=\Huge\boxed{.1}$