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Math competition problem, prove that $\int_{-\infty}^\infty e^{-\pi x^2 \left(\frac{\alpha +x}{\beta +x}\right)^2}dx=1~$ for $~0<\beta<\alpha$.

Other integral related to Ahmed's integral

Indefinite integral $\int \arcsin \left(k\sin x\right) dx$

Proof of a few equations involving $\int_{\alpha}^{\infty}\frac{1}{t\left(e^{t}\pm1\right)}dt$

Is there a closed form for the sum of the cubes of the binomial coefficients?

Finding this summation: $\sum_{n=1}^{\infty}\frac{(2n+99)!(3n-2)!}{(2n)!(3n+99)!}$

Sum: $\sum_{n=1}^\infty\prod_{k=1}^n\frac{k}{k+a}=\frac{1}{a-1}$

Evaluate $\int_{0}^{\frac{\pi}{2}}\frac{dx}{\left(\sqrt{\sin x}+\sqrt{\cos x}\right)^2}$

Another math contest problem: $\int_0^{\frac{\ln^22}4}\,\frac{\arccos\frac{\exp\sqrt x}{\sqrt2}}{1-\exp\sqrt{4\,x}}dx$

Closed-form of $\int_0^{\pi/2}\frac{\sin^2x\arctan\left(\cos^2x\right)}{\sin^4x+\cos^4x}\,dx$

How to evaluate the integral $\int_0^{\pi/2}x^2(\sin x+\cos x)^3\sqrt{\sin x\cos x} \, dx$?

closed form for $\int_0^{\infty}\log^n\left(\frac{e^x}{e^x-1}\right)dx$

How to solve $\int_0^{\infty} \frac{\log(x+\frac{1}{x})}{1+x^2}dx$?

Exact expression of a trigonometric integral

Closed form of $x_{n+1} =\frac{1}{2}\left(x_n-\frac{1}{x_n}\right)$ with $x_0 \neq 0,1$

Evaluate $\int_0^1\frac{\ln(1-x)\ln(1+x)}{1+x^2}dx$

Integral $\int_0^\frac{\pi}{2} \arcsin(\sqrt{\sin x}) dx$

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Is there a closed a form for $\sum_{n=1}^\infty\frac{(-1)^{n-1}H_{an}}{n}\,?$

What is the mathematical relevance of whether an expression has a closed form?

How to calculate $\int_0^\pi \ln(1+\sin x)\mathrm dx$

Math competition problem, prove that $\int_{-\infty}^\infty e^{-\pi x^2 \left(\frac{\alpha +x}{\beta +x}\right)^2}dx=1~$ for $~0<\beta<\alpha$.

Other integral related to Ahmed's integral

Indefinite integral $\int \arcsin \left(k\sin x\right) dx$

Proof of a few equations involving $\int_{\alpha}^{\infty}\frac{1}{t\left(e^{t}\pm1\right)}dt$

Is there a closed form for the sum of the cubes of the binomial coefficients?

Finding this summation: $\sum_{n=1}^{\infty}\frac{(2n+99)!(3n-2)!}{(2n)!(3n+99)!}$

Sum: $\sum_{n=1}^\infty\prod_{k=1}^n\frac{k}{k+a}=\frac{1}{a-1}$

Evaluate $\int_{0}^{\frac{\pi}{2}}\frac{dx}{\left(\sqrt{\sin x}+\sqrt{\cos x}\right)^2}$

Another math contest problem: $\int_0^{\frac{\ln^22}4}\,\frac{\arccos\frac{\exp\sqrt x}{\sqrt2}}{1-\exp\sqrt{4\,x}}dx$

Closed-form of $\int_0^{\pi/2}\frac{\sin^2x\arctan\left(\cos^2x\right)}{\sin^4x+\cos^4x}\,dx$

How to evaluate the integral $\int_0^{\pi/2}x^2(\sin x+\cos x)^3\sqrt{\sin x\cos x} \, dx$?

closed form for $\int_0^{\infty}\log^n\left(\frac{e^x}{e^x-1}\right)dx$

How to solve $\int_0^{\infty} \frac{\log(x+\frac{1}{x})}{1+x^2}dx$?

Exact expression of a trigonometric integral

Closed form of $x_{n+1} =\frac{1}{2}\left(x_n-\frac{1}{x_n}\right)$ with $x_0 \neq 0,1$

Evaluate $\int_0^1\frac{\ln(1-x)\ln(1+x)}{1+x^2}dx$

Integral $\int_0^\frac{\pi}{2} \arcsin(\sqrt{\sin x}) dx$