Suppose we want to show that $$n! \sim \sqrt{2 \pi} n^{n+(1/2)}e^{-n}$$ Instead we could show that $$\lim_{n \to \infty} \frac{n!}{n^{n+(1/2)}e^{-n}} = C$$ where $C$ is a constant. Maybe $C = \sqrt{... 4answers 3k views ### A conjectured closed form of$\int\limits_0^\infty\frac{x-1}{\sqrt{2^x-1}\ \ln\left(2^x-1\right)}dx$Consider the following integral: $$\mathcal{I}=\int\limits_0^\infty\frac{x-1}{\sqrt{2^x-1}\ \ln\left(2^x-1\right)}dx.$$ I tried to evaluate$\mathcal{I}$in a closed form (both manually and using ... 3answers 3k views ### A nasty integral of a rational function I'm having a hard time proving the following $$\int_0^{\infty} \frac{x^8 - 4x^6 + 9x^4 - 5x^2 + 1}{x^{12} - 10 x^{10} + 37x^8 - 42x^6 + 26x^4 - 8x^2 + 1} \, dx = \frac{\pi}{2}.$$ Mathematica has no ... 4answers 1k views ### Integral${\large\int}_0^\infty\frac{dx}{\sqrt[4]{7+\cosh x}}$How to prove the following conjectured identity? $$\int_0^\infty\frac{dx}{\sqrt[4]{7+\cosh x}}\stackrel{\color{#a0a0a0}?}=\frac{\sqrt[4]6}{3\sqrt\pi}\Gamma^2\big(\tfrac14\big)\tag1$$ It holds ... 1answer 991 views ### Prove${\large\int}_0^\infty\frac{\ln x}{\sqrt{x}\ \sqrt{x+1}\ \sqrt{2x+1}}dx\stackrel?=\frac{\pi^{3/2}\,\ln2}{2^{3/2}\Gamma^2\left(\tfrac34\right)}$I discovered the following conjecture by evaluating the integral numerically and then using some inverse symbolic calculation methods to find a possible closed form: $$\int_0^\infty\frac{\ln x}{\sqrt{... 3answers 623 views ### Closed-form of \int_0^\infty \frac{1}{\left(a+\cosh x\right)^{1/n}} \, dx for a=0,1 While I was working on this question by @Vladimir Reshetnikov, I've conjectured the following closed-forms.$$ I_0(n)=\int_0^\infty \frac{1}{\left(\cosh x\right)^{1/n}} \, dx \stackrel{?}{=} \frac{\... 5answers 677 views ### Compute$\int_0^{\pi/2}\frac{\cos{x}}{2-\sin{2x}}dx$How can I evaluate the following integral? $$I=\int_0^{\pi/2}\frac{\cos{x}}{2-\sin{2x}}dx$$ I tried it with Wolfram Alpha, it gave me a numerical solution:$0.785398$. Although I immediately ... 4answers 572 views ### Prove that$\int_0^4 \frac{\ln x}{\sqrt{4x-x^2}}~dx=0$(without trigonometric substitution) The integral is from P. Nahin's "Inside Interesting Integrals...", problem C2.1. His proposed solution includes trigonometric substitution and the use of log-sine integral. However, I think the ... 3answers 194 views ### Calculate an integral using complex integration came across this one $$\int_0^{\pi / 2} \ln (\sin x)\;dx$$ I wanted to find it using the residues, but, I don't thing they are isolated ones 2answers 156 views ### how to regularize$ \int_{-\infty}^\infty x \sin x \, dx \$? [closed]
In a physics problem I am confronted with a divergent integral $$\int_{-\infty}^\infty x \sin x \, dx = \sin x - x \cos x \bigg|_{-\infty}^\infty \approx 0$$ How to regularize it? in ...