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Let $a<b$ be a positive real numbers. Are there tight upper bounds for $\int_a^{b} \frac{\exp(x)}{x}\ dx$, specially asymptotic bounds when $a, b,\frac{b}{a}\to\infty$?

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Applying integration by parts $n$ times gives

\begin{align*} \int_{a}^{b} \frac{e^x}{x} \, \mathrm{d}x = e^b R_n(b) - e^a R_n(a) + \int_{a}^{b} \frac{n! e^{x}}{x^{n+1}} \, \mathrm{d}x, \end{align*}

where $R_n(x) = \sum_{k=1}^{n} \frac{(k-1)!}{x^k}$. Now, as $a, b, (b/a) \to \infty $, the last integral is bounded by $\mathcal{O}(e^b / b^{n+1})$, and so, we get

$$ \int_{a}^{b} \frac{e^x}{x} \, \mathrm{d}x = e^b R_{n}(b) + \mathcal{O}(e^b /b^{n+1}). $$

as $a, b, (b/a) \to \infty$, for each fixed $n \geq 1$.

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The integral will be asymptotically $\frac{e^b}b$.

The lower bound $$ \int_a^b \frac{e^x}x\,dx > \int_a^b \frac{e^x}b\,dx = \frac{e^b-e^a}b $$ is easy. For an upper bound, for any $c\in(a,b)$ we have \begin{align*} \int_a^b \frac{e^x}x\,dx &= \int_a^c \frac{e^x}x\,dx + \int_c^b \frac{e^x}x\,dx \\ &< \int_a^c \frac{e^x}a\,dx + \int_c^b \frac{e^x}c\,dx \\ &= \frac{e^c-e^a}a + \frac{e^b-e^c}c; \end{align*} taking $c$ close (but not too close) to $b$, for example $c=b-\sqrt{ab}$ (valid when $\frac ba$ isn't too small), recovers the asymptotic $\frac{e^b}b$.

One takeaway: always try trivial bounds first; and if the integrand (or summand) consists of an increasing function times a decreasing function, try cutting the integral (or series) at an arbitrary middle point and try trivial bounds on each portion separately.

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