# Linked Questions

3answers
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### 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 $$\lim_{a\to\infty} \int_0^a e^{-xt}\, dt = \frac{1}{x}, x\gt 0,$$ ...
1answer
880 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 ...
1answer
1k 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 ...
1answer
270 views

6answers
3k 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 ...
9answers
7k views

I am looking for a short proof that $$\int_0^\infty \left(\frac{\sin x}{x}\right)^2 \mathrm dx=\frac{\pi}{2}.$$ What do you think? It is kind of amazing that $$\int_0^\infty \frac{\sin x}{x} \mathrm ... 6answers 9k views ### Does  \int_0^{\infty}\frac{\sin x}{x}dx  have an improper Riemann integral or a Lebesgue integral? In this wikipedia article for improper integral,$$ \int_0^{\infty}\frac{\sin x}{x}dx $$is given as an example for the integrals that have an improper Riemann integral but do not have a (proper) ... 6answers 732 views ### show that \int_{0}^{\infty} \frac {\sin^3(x)}{x^3}dx=\frac{3\pi}{8} show that$$\int_{0}^{\infty} \frac {\sin^3(x)}{x^3}dx=\frac{3\pi}{8}$$using different ways thanks for all 2answers 500 views ### Evaluating \int_0^{\infty} \text{sinc}^m(x) dx How do I evaluate$$I_m = \displaystyle \int_0^{\infty} \text{sinc}^m(x) dx,$$where m \in \mathbb{Z}^+? For m=1 and m=2, we have the well-known result that this equals \dfrac{\pi}2. In ... 3answers 372 views ### How to find \int\frac{\sin x}{x}dx How do I integrate$$\int\frac{\sin(x)}xdx? I tried using integration by parts, but it led me to nowhere. Please help.

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