Linked Questions

2
votes
3answers
573 views

Evaluating $\lim_{b\to\infty} \int_0^b \frac{\sin x}{x}\, dx= \frac{\pi}{2}$ [duplicate]

Possible Duplicate: Solving the integral $\int_{0}^{\infty} \frac{\sin{x}}{x} \ dx = \frac{\pi}{2}$? Using the identity $$\lim_{a\to\infty} \int_0^a e^{-xt}\, dt = \frac{1}{x}, x\gt 0,$$ ...
1
vote
1answer
569 views

how to integrate $ \int_{-\infty}^{+\infty} \frac{\sin(x)}{x} \,dx $? [duplicate]

Possible Duplicate: Solving the integral $\int_{0}^{\infty} \frac{\sin{x}}{x} \ dx = \frac{\pi}{2}$? How can I do this integration using only calculus? (not laplace transforms or complex ...
0
votes
1answer
596 views

Dirichlet integral. [duplicate]

I want to prove $\displaystyle\int_0^{\infty} \frac{\sin x}x dx = \frac \pi 2$, and $\displaystyle\int_0^{\infty} \frac{|\sin x|}x dx \to \infty$. And I found in wikipedia, but I don't know, can't ...
-1
votes
1answer
349 views

How to prove $\int^{\infty}_{0}\frac{\sin x}{x}\mathrm{d}x=\frac{\pi}{2}$ [duplicate]

Possible Duplicate: Solving the integral $\int_{0}^{\infty} \frac{\sin{x}}{x} \ dx = \frac{\pi}{2}$? How to prove $$\int^{\infty}_{0}\frac{\sin x}{x}\mathrm{d}x=\frac{\pi}{2}$$ Do you have ...
1
vote
1answer
162 views

How to prove $\operatorname{si}(0) = -\pi/2$ without contour [duplicate]

Possible Duplicate: Solving the integral $\int_{0}^{\infty} \frac{\sin{x}}{x} \ dx = \frac{\pi}{2}$? How to prove $\operatorname{si}(0) = -\pi/2$ without contour integration ? Where ...
3
votes
1answer
206 views

A 'complicated' integral: $ \int \limits_{-\infty}^{\infty}\frac{\sin(x)}{x}$ [duplicate]

I am calculating an integral $\displaystyle \int \limits_{-\infty}^{\infty}\dfrac{\sin(x)}{x}$ and I dont seem to be getting an answer. When I integrate by parts twice, I get: $$\displaystyle \int ...
0
votes
1answer
125 views

How can I evaluate $\int_0^\infty \frac{\sin x}{x} \,dx$? [may be duplicated] [duplicate]

How can I evaluate $\displaystyle\int_0^\infty \frac{\sin x}{x} \, dx$? (Let $\displaystyle \frac{\sin0}{0}=1$.) I proved that this integral exists by Cauchy's sequence. However I can't evaluate ...
0
votes
0answers
33 views

Show that $lim_{R\rightarrow \infty}\int^R_0 \frac{\sin x}{x} dx= \frac{\pi}{2}$. [duplicate]

I came across this qualifying exam problem and wasn't sure what to do. Using techniques of real analysis (as opposed to complex analysis) show that $lim_{R\rightarrow \infty}\int^R_0 \frac{\sin x}{x} ...
42
votes
6answers
2k views

How to prove this identity $\pi=\sum\limits_{k=-\infty}^{\infty}\left(\frac{\sin(k)}{k}\right)^{2}\;$?

How to prove this identity? $$\pi=\sum_{k=-\infty}^{\infty}\left(\dfrac{\sin(k)}{k}\right)^{2}\;$$ I found the above interesting identity in the book $\bf \pi$ Unleashed. Does anyone knows how to ...
30
votes
9answers
5k views

Proof of $\int_0^\infty \left(\frac{\sin x}{x}\right)^2 dx=\frac{\pi}{2}.$

I am looking for a short proof that $$\int_0^\infty \left(\frac{\sin x}{x}\right)^2 dx=\frac{\pi}{2}.$$ What do you think? It is kind of amazing that $$\int_0^\infty \frac{\sin x}{x} dx$$ is also ...
23
votes
6answers
5k views

Does $ \int_0^{\infty}\frac{\sin x}{x}dx $ have an improper Riemann integral or a Lebesgue integral?

In this wikipedia article for improper integral, $$ \int_0^{\infty}\frac{\sin x}{x}dx $$ is given as an example for the integrals that have an improper Riemann integral but do not have a (proper) ...
5
votes
5answers
501 views

show that $\int_{0}^{\infty} \frac {\sin^3(x)}{x^3}dx=\frac{3\pi}{8}$

show that $$\int_{0}^{\infty} \frac {\sin^3(x)}{x^3}dx=\frac{3\pi}{8}$$ using different ways thanks for all
17
votes
2answers
360 views

Evaluating $\int_0^{\infty} \text{sinc}^m(x) dx$

How do I evaluate $$I_m = \displaystyle \int_0^{\infty} \text{sinc}^m(x) dx,$$ where $m \in \mathbb{Z}^+$? For $m=1$ and $m=2$, we have the well-known result that this equals $\dfrac{\pi}2$. In ...
5
votes
3answers
268 views

Integration doubt

How do I integrate $\sin(x)/x$? I tried using integration by parts, but it led me to nowhere. Please help.

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