The notation $J_0\left(x\right)$ represents the special function Maple knows as BesselJ. The full syntax for this function would be $\mathrm{BesselJ}\left(0\,x\right)$, but by invoking special typesetting rules through the Typesetting Rules Assistant, (View menu, Typesetting Rules) Maple can access this Bessel function with the simpler syntax $J_0\left(x\right)$.

Maple evaluates the given integral numerically, giving ${\int }_0^{1.0}{J}_0\left(x\right) \ⅆx\doteq 0.9197304101$. Expanding the integrand in a Maclaurin series gives

$J_0\left(x\right)\=1-\frac{1}{4}{x}^2\+\frac{1}{64}{x}^4-\frac{1}{2304}{x}^6\+\frac{1}{147456}{x}^8\+\cdots$$$

The results of termwise integration of the series expansion are listed in Table 8.5.12(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 8.5.12(a)$$ suggests that just the first three terms need to be added for the approximation to be accurate to three decimal places.