11k views

### Intuition for the definition of the Gamma function?

In these notes by Terence Tao is a proof of Stirling's formula. I really like most of it, but at a crucial step he uses the integral identity $$n! = \int_{0}^{\infty} t^n e^{-t} dt$$ coming from ...
6k views

### Find the average of $\sin^{100} (x)$ in 5 minutes?

I read this quote attributed to VI Arnold. "Who can't calculate the average value of the one hundredth power of the sine function within five minutes, doesn't understand mathematics - even if he ...
5k views

### Factorial and exponential dual identities

There are two identities that have a seemingly dual correspondence: $$e^x = \sum_{n\ge0} {x^n\over n!}$$ and $$n! = \int_0^{\infty} {x^n\over e^x}\ dx.$$ Is there anything to this comparison? (I ...
7k views

### Computing the integral of $\log(\sin x)$

How to compute the following integral? $$\int\log(\sin x)\,dx$$ Motivation: Since $\log(\sin x)'=\cot x$, the antiderivative $\int\log(\sin x)\,dx$ has the nice property $F''(x)=\cot x$. Can we ...
7k views

### Purpose Of Adding A Constant After Integrating A Function

I would like to know the whole purpose of adding a constant termed constant of integration everytime we integrate an indefinite integral $\int f(x)dx$. I am aware that this constant "goes away" when ...
2k views

I recently showed my Algorithms class how to bound $\ln n! = \sum \ln n$ by integrals, thereby obtaining the simple factorial approximation $$e \left(\frac{n}{e}\right)^{n} \leq n! \leq en\left(\... 3answers 3k views ### Showing that \int_0^1 \log(\sin \pi x)dx=-\log2 I need help with a textbook exercise (Stein's Complex Analysis, Chapter 3, Exercises 9). This exercise requires me to show that$$\int_0^1 \log(\sin \pi x)dx=-\log2$$A hint is given as "Use the ... 3answers 583 views ### How to solve integral recursive relation - I_n=\int_0^1(x-x^2)^ndx Let I_n=\int_0^1(x-x^2)^ndx. Prove that I_n=\frac{1}{4}\cdot\frac{2n}{2n+1}I_{n-1}. This sounds like a rather easy exercise, but no matter how hard I try, I can't quite put my finger on it (I ... 3answers 499 views ### Closed form for integral \int_{0}^{\pi} \left[1 - r \cos\left(\phi\right)\right]^{-n} \phi \,{\rm d}\phi Is there a closed form for$$I_n =\int_{0}^{\pi} \frac{\phi}{(1 - r \cos\phi)^n} \,{\rm d}\phi $$for \left\vert\,r\,\right\vert < 1 real and n > 0 integer ? The solution to this integral ... 4answers 134 views ### Evaluate the integral \int^{\infty}_{0} e^{-x}x^{100}dx$$\int^{\infty}_{0} e^{-x}x^{100}dx$$I am sure is something here I can not see, else it is integration by parts 100 times. 3answers 891 views ### How to find \int_0^{\infty}\frac{dx}{(1+x^2)^4} How would you compute for the definite integral of$$\int_0^{\infty}\frac{dx}{(1+x^2)^4}$$I know that integral of \displaystyle \frac1{(1+x^2)} equals \tan^{-1}x. I tried using integration by ... 4answers 200 views ### Practice Preliminary exam - evaluate the limit This is from a practice prelim exam and I know I should be able to get this one.$$ \lim_{n\to\infty} n^{1/2}\int_0^\infty \left( \frac{2x}{1+x^2} \right)^n $$I have tried many different u-... 4answers 202 views ### How to prove \lim_{n \to +\infty} \sqrt{n}\int_0^\pi{\cos(\frac{t}{2})^n}dt>0 I want to prove$$\lim_{n \to +\infty}\sqrt{n}\int_0^\pi{\cos\left(\frac{t}{2}\right)^n}dt>0.$$First, I consider$$\lim_{n \to +\infty}\sqrt{n}\int_0^\pi{\cos\left(\frac{t}{2}\right)^n}\sin\left(\...
I want to know the answer/references for the question on decay of Fourier series coefficients and the differentiability of a function. Does the magitude of fourier series coefficients {$a_k$} of a ...
I have this integral: $$\int_{-1}^{1} \frac{e^x}{\sqrt{1-x^2}}\,dx$$ How can I get rid of the infinities at the ends of the interval so that I can evaluate this integral numerically? I tried to make ...