So I correctly found the MacLaurin series for $\int \sin(x^2)dx$, but I don't know what to plug into the polynomial to estimate the integral from $0$ to $1$. I don't think I do one set of the polynomial at 1, and another at 0, because I tried that and the answer does not match the key.
EDIT: It tells me to use the first 3 terms of my series (these terms have already been integrated): $x-\frac{x^7}{7*3!}+\frac{x^{11}}{11*5!}$
$f(x) = sin(x) = \sum_{i=0}^n(((-1)^n/(2n+1)!)x^n $ Taylor Polynome for f(x) = sin(x) Taylor Expansion for $f(x)=sin(x)$
if $f(u) = sin(u), u = x^2, then, f (x^2) =\sum_{i=0}^n(((-1)^n/(2n+1)!)(x^2)^n $
Integrating:
$\int_{0}^1 f(x^2) dx$ $\int_{0}^1 \sum_{i=0}^n(((-1)^n/(2n+1)!)(x^2)^n dx$
Expand the polynomial expression whatever do you want, integrate it and then evaluate from 0 to 1.