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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 $$\lim_{a\to\infty} \int_0^a e^{-xt}\, dt = \frac{1}{x}, x\gt 0,$$ ...
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### 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 ...
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 ...
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### 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 ...