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### Dirichlet integral. [duplicate]

I want to prove $\displaystyle\int_0^{\infty} \frac{\sin x}x \,\mathrm{d}x = \frac \pi 2$, and $\displaystyle\int_0^{\infty} \frac{|\sin x|}x \,\mathrm{d}x \to \infty$. And I found in wikipedia, but ...
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### 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 ...
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### Alternative approach to the computation of $\int_0^\infty\frac{\sin(nx)}xdx$ [duplicate]

I am not able to obtain the value of:$$\int_{0}^{\infty}\dfrac{\sin(nx)}{x}dx ,$$ By changing the order of integration of: $$\int_{0}^{\infty}\int_{0}^{\infty}e^{-xy}\sin(nx) dx dy.$$ And by using ...
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### 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 a ...
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### Integration of $\sin x/x$ [duplicate]

How to evaluate $\displaystyle \int_0^\infty \frac{\sin x} x \, dx$ I have tried substitutions and integration by parts, but unable to integrate it.
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### How to prove $\int_0^{\infty} \frac{\sin x}{x} \, dx = \frac{\pi}{2}$ [duplicate]
How to prove $\displaystyle \int_0^{\infty} \frac{\sin x}{x} \, dx = \frac{\pi}{2}$? How do I proceed?
### Compute $\int_0^\infty \frac{\sin(x)}{x}dx$ without residue theorem. [duplicate]
Is there a way to compute $$\int_0^\infty \frac{\sin(x)}{x}dx$$ without residue theorem ?