# $\int_0^{\infty} \frac{e^{-\pi x^2}+e^{-\pi /x^2}}{e^{\pi x}+e^{\pi /x}} dx$ Sum of Coefficients in Cubic Polynomial

I have been trying to solve this problem and I am stuck half way.

My Working:

I have solved the integrals and noticed that the value of $$\alpha$$ comes out to be $$\alpha\approx 94.05$$ So, the cubic equation becomes $$94.05^3A+94.05^2B+94.05C+D=0$$ I am wondering if the question is correct or not. Because I think we cannot solve for four unknowns $$-(A+B+C+D)= ?$$ using 1 equation only, I think we need three more equations. There may be many possible answers for this equation as coefficients are not defined. If it is still solvable please let me know, $$\color{red}{\text{I am not asking answer I just want to know is it solvable or not ?}}$$

Thanks

Edit : $$\color{blue}{\text{The original problem has been deleted now}}$$ on the site maybe due to wrong question statement and not properly stated or something is missing. Maybe they feel it is not possible to find the value of $$A+B+C+D$$

However if you still want to try these lovely integrals then you can. Thank you for the support.

• How did you find $\alpha$? Commented May 12, 2021 at 8:11
• So these integrals are solvable? Could you write up a brief sketch of the solution? Commented May 12, 2021 at 8:21
• Scaling $A,B,C,D$ works, so there isn't any unique answer. Where is this problem from? Commented May 12, 2021 at 10:11
• from brilliant,org this is the link brilliant.org/problems/tale-of-2-exponential-integrals Commented May 12, 2021 at 11:52
• It seems that $A, B, C, D$ are integers, since hint says that the answer is a prime number. However, I can't figure out why $\alpha$ should be the root of integer coefficient cubic polynomial (so is algebraic). Commented May 13, 2021 at 2:52