Evaluating $\int_0^{\infty}\frac{e^{-x}}{1+x^2}dx$

I'm trying to evaluate $$\int_0^{\infty}\dfrac{e^{-x}}{1+x^2}dx$$ By making the substitution $x=\tan\theta$, $$\int_0^{\infty}\dfrac{e^{-x}}{1+x^2}dx=\int_0^{\frac \pi 2}\exp(-\tan\theta)d\theta$$ So it converges to something less than $\frac \pi 2$. Is there any way to find the exact value, using only elementary methods?

• You can evaluate the integral in terms of the exponential integral. – Mhenni Benghorbal Apr 3 '13 at 12:33
• @IshanBanerjee: Not every integral can be evaluated in terms of elementary functions. – Mhenni Benghorbal Apr 3 '13 at 12:38
• The indefinite integral isn't elementary , but is it possible to do one with these limits without involving the exponential integral? – Ishan Banerjee Apr 3 '13 at 12:41
• Nope. Look up the Laplace transform of $\frac1{1+x^2}$ in any convenient table (or evaluate with any software at hand), and report back. – J. M. is a poor mathematician Apr 3 '13 at 12:42
• @IshanBanerjee: Why should you delete it? Just leave it. – Mhenni Benghorbal Apr 3 '13 at 12:58

As pointed out in the comments and in Xiaolang's answer, this integral is very unlikely to have a closed-form solution in terms of elementary functions, but it is possible to express it in terms of exponential integrals or, equivalently, sine and cosine integrals. These hardly qualify as "elementary methods", but we can at least eliminate the sine and cosine integrals using their power series expansions.

We start with the representation in terms of the cosine and sine integrals $\operatorname{Ci}$ and $\operatorname{Si}$: \begin{align} \int_0^\infty \frac{e^{-x}}{1+x^2} &= \operatorname{Ci}(1)\sin(1) - \operatorname{Si}(1)\cos(1)+\frac{\pi \cos(1)}{2},\\\ \\ \operatorname{Ci}(x) &= \gamma + \ln(x) +\int_0^x \frac{\cos(t)-1}{t}dt, \\\ \\ \operatorname{Si}(x) &= \int_0^x \frac{\sin(t)}{t}dt, \end{align} where $\gamma=0.577\ldots$ is the Euler-Mascheroni constant. Power series expansions for $\operatorname{Ci}(x)$ and $\operatorname{Si}(x)$ follow directly from their definitions in terms of sines and cosines: \begin{align} \operatorname{Ci}(x) &= \gamma + \ln(x) + \sum_{j=0}^\infty \frac{(-1)^{j+1}x^{2j+2}}{(2j+2)!(2j+2)},\\\ \\ \operatorname{Si}(x) &= \sum_{j=0}^\infty \frac{(-1)^jx^{2j+1}}{(2j+1)!(2j+1)}. \end{align} Substituting $x=1$ into these expressions, applying the power series expansions for $\sin(1)$ and $\cos(1)$ where they multiply $\operatorname{Ci}(1)$ and $\operatorname{Si}(1)$, and doing a somewhat non-negligible amount of arithmetic, we find: $$\int_0^\infty \frac{e^{-x}}{1+x^2} = \gamma \sin(1) + \frac{\pi\cos(1)}{2} + \sum_{k=0}^\infty \sum_{j=1}^\infty \frac{f(j-k)}{j(j!)(k!)}. \tag{\star}$$ Here the function $f(j-k)$ is the sequence $0,-1,0,1,\ldots$: \begin{align} f(j-k) &= 0\ \ \quad \textrm{if}\quad j-k\equiv 0\ (\operatorname{mod} 4)\\ &= -1\ \ \ \,\textrm{if}\quad j-k\equiv 1\ (\operatorname{mod} 4)\\ &= 0\ \ \quad \textrm{if}\quad j-k\equiv 2\ (\operatorname{mod} 4)\\ &= 1\ \ \quad \textrm{if}\quad j-k\equiv 3\ (\operatorname{mod} 4).\\ \end{align} Clearly Eq. $(\star)$ converges rapidly, considering the factorial functions in the denominator of the summand. Indeed, including just 25 terms (5 values for each index), we obtain 0.621..., consistent with the result 0.621449... quoted previously.

• PS. In your question you quote $\pi/2$ as an upper bound. Actually, note that we have $1/(x^2+1)\leq 1\ \forall\ x\in\mathbb{R}$, with the equality only at $x=0$. This gives: $$\int_0^\infty \frac{e^{-x}}{{x^2+1}}dx < 1=\int_0^\infty e^{-x}dx.$$ – Douglas B. Staple Apr 12 '13 at 2:58
• Though we didn't get an elementary answer, your post provided some additional insight. Congratulations to the +100. – Mårten W Apr 12 '13 at 11:08
• @MårtenW Thanks. I found this problem challenging and had fun looking at it. It's a shame that the double sum in $(\star)$ couldn't be identified with any known constants. I'd be interested to hear if someone manages to do better. – Douglas B. Staple Apr 14 '13 at 2:43

there may not be a elementray functions but it isn't a easy thing to prove you can consult The Liouville's therom in complex analysis

$$\int_0^{\pi/2}e^{\tan(\theta)}d\theta= \operatorname{Ci}(1)\sin(1)- \operatorname{Si}(1)\cos(1)+\pi \cos(1)/2 \approx 0.62144962423581335763926$$