Linked Questions

53
votes
9answers
11k views
+100

Stirling's formula: proof?

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{...
43
votes
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 ...
60
votes
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 ...
34
votes
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 ...
35
votes
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{...
15
votes
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{\...
9
votes
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 ...
7
votes
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 ...
4
votes
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
3
votes
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 ...