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in the Classical Mechanics (2nd. Ed.) book of Herbert Goldstein, p. 75 it says: "Equations 3-18 and 3-20 are the two remaining integrations, and formally the problem has been reduced to quadratures..."

It is not clear to me what a quadrature is, is it simply the integration process through some numerical analysis method such as trapezoidal rule? Or is it the process of reducing the order of the initial partial differential equation by using constants of motion?

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    $\begingroup$ Quadrature : In mathematics, quadrature is a historical term which means determining area. This term is still used nowadays in the context of differential equations, where "solving an equation by quadrature" means expressing its solution in terms of integrals. $\endgroup$ – Frobenius Sep 19 '18 at 18:47
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As Frobenius mentioned in comment, quadrature is a historic term for integration.

But in modern texts I tend to only see quadrature used in the context of numeric approximation methods, in order to avoid confusing those methods with exact mathematical methods of integration.

In particular quadrature nowadays seems to refer to integration over an explicit range, whereas integration is the inverse of differentiation and does not require a range. See also Antiderivative.

So for example :

$$\int_a^by(x)dx \approx \sum_{n=1}^{N}\,y\left(a+\frac n N (b-a)\right)\frac{b-a}N \tag{a quadrature}$$

and :

$$\int_0^{2\pi}sin^2(\theta)d\theta = \pi \tag{an exact integral}$$

and :

$$\int sin^2(\theta)d\theta = \frac \theta 2 - \frac 1 4 sin(2\theta) \tag{an indefinite integral}$$

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