calculus - how to integrate $ int_{-infty}^{+infty} frac{sin(x)}{x} ,dx $?

How can I do this integration using only calculus?
(not laplace transforms or complex analysis)

$$
\int_{-\infty}^{+\infty} \frac{\sin(x)}{x} \,dx
$$

I searched for solutions not involving laplace transforms or complex analysis but I could not find.

Answer

Putting rigor aside, we may do like this:
$$\begin{align*}
\int_{-\infty}^{\infty} \frac{\sin x}{x} \; dx

&= 2 \int_{0}^{\infty} \frac{\sin x}{x} \; dx \\
&= 2 \int_{0}^{\infty} \sin x \left( \int_{0}^{\infty} e^{-xt} \; dt \right) \; dx \\
&= 2 \int_{0}^{\infty} \int_{0}^{\infty} \sin x \, e^{-tx} \; dx dt \\
&= 2 \int_{0}^{\infty} \frac{dt}{t^2 + 1} \\
&= \vphantom{\int}2 \cdot \frac{\pi}{2} = \pi.
\end{align*}$$
The defects of this approach are as follows:

1. Interchanging the order of two integral needs justification, and in fact this is the hardest step in this proof. (There are several ways to resolve this problem, though not easy.)



2. It is nothing but a disguise of Laplace transform method. So this calculation contains no new information on the integral.

summation - Find the following limit: $limlimits_{n to infty}(e-1)sum_{k=1}^n frac{1}{n+k(e-1)} $

I'm having problems with the following limit:

$\lim\limits_{n \to \infty}(e-1)\sum_{k=1}^n \frac{1}{n+k(e-1)} $

It's a task from a taskbook for first year engineering students. Any help is appreciated. Thanks.

calculus - Calculate the Gauss integral without squaring it first

We know that the integral
$$I = \int_{-\infty}^{\infty} \mathrm{d}x e^{-x^2}$$
can be calculated by first squaring it and then treat it as a $2-$dimensional integral in the plane and integrate it in polar coordinates.

Are there any other ways to calculate it? I know that we may use the relation
$$\Gamma(x)\Gamma(1-x) = \frac{\pi}{\sin{\pi x}},$$
but this, in effect, is still taking the square.

Well, after I write down the above text, I figure that maybe there is no way to calculate it without squaring, since, after all, the result contains a square root, and it seems no elementary function can "naturally" produce a square root of $\pi$ starting from natural numbers (though I don't know how to describe this more concretely; you are also welcome to comment on this point). Nevertheless I still post this question in case there are some other ideas.

EDIT: the Fourier transformation method at Computing the Gaussian integral with Fourier methods? appears kind of cheat to me, since the very proof of the Fourier transformation formula actually makes use of the value of the Gauss integral (at least in this wiki page http://en.wikipedia.org/wiki/Fourier_inversion_theorem#Proof).

Thank you.

summation - How to find the consecutive odd numbers that sum to a given odd number

Given a non perfect square odd number, say $1649$ What is the most efficient way to find the consecutive odd positive integers that sum to that number.

No other information is provided, just the odd number to solve.

In reality the odd numbers I want to find are substantially larger than 1649 which would be trivial to find the consecutive odds that sum to it. So I am looking for an efficient generalised method to turn into an algorithm to resolve this problem for any given odd number.

EDIT:
Perhaps I should have added. The numbers I am working with are too large to factor in an efficient manner. Therefore the difference of two squares is not a workable option.

I am looking for alternate approaches to solving the problem.

real analysis - An alternative way to find the sum of this series?

$\displaystyle \frac{4}{20}$+$\displaystyle \frac{4.7}{20.30}$+$\displaystyle \frac{4.7.10}{20.30.40}$+...

Now I have tried to solve this in a usual way, first find the nth term $t_n$.

$t_n$= $\displaystyle \frac{1}{10}$($\displaystyle \frac{1+3}{2}$) + $\displaystyle \frac{1}{10^2}$($\displaystyle \frac{1+3}{2}$)($\displaystyle \frac{1+6}{3}$) + ...+ $\displaystyle \frac{1}{10^n}$($\displaystyle \frac{1+3}{2}$)($\displaystyle \frac{1+6}{3}$)...($\displaystyle \frac{1+3n}{n+1}$)

=$\displaystyle \frac{1}{10^n}\prod$(1+$\displaystyle \frac{2r}{r+1}$) , $r=1,2,..,n$

=$\displaystyle \prod$($\displaystyle \frac{3}{10}-\displaystyle \frac{1}{5(r+1)}$)
thus, $t_n=$ (x-$\displaystyle \frac{a}{2}$)(x-$\displaystyle \frac{a}{3}$)...(x-$\displaystyle \frac{a}{n+1}$), x=$\displaystyle \frac{3}{10}$, a=$\displaystyle \frac{1}{5}$

Now to calculate $S_n$, I have to find the product $t_n$, and then take sum over it. But this seems to be a very tedious job. Is there any elegant method(may be using the expansions of any analytic functions) to do this?

gcd and lcm - using extended euclidean algorithm to find s, t, r

i am stuck for many hours and i don't understand using the extended euclidean algorithm. i calculated it the gcd using the regular algorithm but i don't get how to calculate it properly to obtain s,t,r.

i understand that from the gcd i can get a linear combination representation, but i don't get how to do it using the algorithm.

how can i find $s,t,r$ for $a=154, b= 84$?

if it is of any importance, the algorithm i am referring to is from the book cryptography: theory and practice

thank you very much. became hopeless because of it

Answer

Using the Euclidean algorithm, we have

$$
\begin{align}
154&=1\cdot84+70\tag{1}\\
84&=1\cdot70+14\tag{2}\\

70&=5\cdot14+0
\end{align}
$$
The last nonzero remainder is 14. So $\gcd(154,84)=14$. Now
$$
\begin{align*}
14&=84-70\qquad\text{(using 2)}\\
&=84-(154-84)\qquad\text{(using 1)}\\
&=2\cdot84-1\cdot154
\end{align*}

$$
So $14=2\cdot84-1\cdot154$.

real analysis - Turning a continuous everywhere differentiable nowhere function into a smooth function by infinitely many times definite integration?

Let $W(x)$ be a real-vlued function defined on a (possibly infinite) interval $\text{T}\subseteq\mathbb{R}$ containing $0$ that is continuous everywhere differentiable nowhere on $\text{T}$.

Define the sequence of function $f_n:\text{T}\to\mathbb{R}$ as follows:
$$f_0(x)\triangleq{W(x)}$$ and $$f_n(x)\triangleq\int_{0}^{x}f_{n-1}(u)du$$ for $n=1,2,3,4,5,...$ .

Then it follows the fact that $f_n$ is differentiable exactly $n$-times everywhere on $\text{T}$.

Question:

Does there exist such $W(x)$ and $\text{T}$ so that the sequence of function $f_n$ on $\text{T}$ "converges" to a limit function $f_{\infty}$ on $\text{T}$ with $f_{\infty}$ being infinitely many times differentiable (i.e., smooth) on $\text{T}$?