Maple can evaluate the given integral exactly in terms of the special function FresnelS. This exact solution can then be approximated to a sufficiently high accuracy.

${\int }_0^1\sin \left(x^2\right) \mathit{\ⅆ}x$ = $\frac{1}{2}\mathrm{FresnelS}\left(\frac{\sqrt{2}}{\sqrt{\mathrm{\π}}}\right)\sqrt{2}\sqrt{\mathrm{\π}}$$\doteq$$0.3102683013$ 

Fresnel is the name of a French physicist. The correct pronounciation of his name can be heard at a number of internet sites, including the one at the end of this link.

Expanding the integrand in a Maclaurin series gives

$\sin \left(x^2\right)\=x^2-\frac{1}{6}{x}^6\+\frac{1}{120}{x}^{10}-\frac{1}{5040}{x}^{14}\+\cdots$

The results of termwise integration of the series expansion are listed in Table 8.5.11(a). Because the resulting series is alternating, the remark in Table 8.2.2 applies, that is, the error in a partial sum is less than the first neglected term. Table 8.5.11(a)$$ suggests that just the first three terms need to be added for the approximation to be accurate to three decimal places.