Numerical method for indefinite integral

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let be the indefinite integral

$$ F(x)= \int_{0}^{x} g(t)dt $$

the integral depends on the parameter 'x'

i can use a linear transformation to turn this integral into

$$ F(x)= \int_{-1}^{1}dtu(x,t) $$

so now the dependence on the parameter 'x' is inside the integral

so my question is could now apply a quadrature formula to evaluate the indefinite integral like this

$$ F(x) \approx \sum_{i} c_{i}w(x_{i}g(x,t_{i}) $$

so I approximate the function $ F(x) $ by a sum of functions which will themselves depend on the parameter 'x'

Where could I find more info about this iterated numerical integration in each variable ?