Does the integral $\displaystyle \int_{0}^{x(t_a)} \frac{(1-e^{-x})}{x} dx$ where $0<x(t_a)$ have an analytical form? Does an approximation exist for this integral? Is it possible to approximate this integral using the Ramanujan method?
2026-04-06 12:53:36.1775480016
How to approximate bounded exponential integral?
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You can expand the integrand as a series and integrate term by term getting $$x - x^2/4 + x^3/18 - x^4/96 + x^5/600 + O(x^6)$$ It will converge everywhere, and quickly for small $x$