Linked Questions

59
votes
9answers
3k 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 ...
104
votes
7answers
6k 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 ...
8
votes
8answers
2k 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 ...
35
votes
3answers
3k 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 ...
15
votes
6answers
3k views

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

How to compute the following integral $$\int\log(\sin x)~dx~?$$
16
votes
5answers
1k views

How best to explain the $\sqrt{2\pi n}$ term in Stirling's?

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 ...
7
votes
3answers
400 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 ...
18
votes
3answers
394 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 ...
9
votes
2answers
954 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 ...
8
votes
4answers
172 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 ...
6
votes
3answers
761 views

Tricky integration?

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 ...
5
votes
4answers
195 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 ...
5
votes
1answer
1k views

Differentiability and decay of magnitude of fourier series coefficients

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 ...
6
votes
2answers
137 views

numerical evaluation of an integral

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 ...
3
votes
4answers
139 views

Computing $\int_0^{\pi\over2} \frac{dx}{1+\sin^2(x)}$?

How would you compute$$\int_0^{\pi\over2} \frac{dx}{1+\sin^2(x)}\, \, ?$$

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