Specifically I want to find the upper bound on the Lagrange Error of using the $4th$ degree polynomial for $f(x)=e^{x^2}$ which is $1+x^2+\frac{x^4}{2}$ to find,
$$\int_0^\frac{1}{2} e^{x^2} \,dx$$
So using the Lagrange Error, what is the upper bound on the error in using,
$$\int_0^\frac{1}{2} 1+x^2+\frac{x^4}{2} \,dx$$ to approximate the first integral given?
It's not a good idea to use directly the function $e^{x^2}$, since it fifth derivative is complicated to work with. Instead of that, work with the exponential function. Note that $x\in\left[0,\frac12\right]\implies x^2\in\left[0,\frac14\right]$. And if $x^2\in\left[0,\frac14\right]$, then$$\left|e^{x^2}-\left(1+x^2+\frac{x^4}2\right)\right|\leqslant\frac{\exp\left(\frac14\right)}{5!}\left(\frac14\right)^5<\frac2{120\times1\,024}$$(I'm using here the fact that $\sqrt[4]e<2$). So$$\left|\int_0^{\frac12}e^{x^2}\,\mathrm dx-\int_0^{\frac12}1+x^2+\frac{x^4}2\,\mathrm dx\right|<\frac1{120\times1\,024}.$$