# Linked Questions

5answers
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### Are there any ways to evaluate $\int^\infty_0\frac{\sin x}{x}dx$ without using double integral? [duplicate]

Are there any ways to evaluate $\int^\infty_0\frac{\sin x}{x}dx$ without using double integral? I can't find any this kind of solution. Can anyone please help me? Thank you.
3answers
1k 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 $$\lim_{a\to\infty} \int_0^a e^{-xt}\, dt = \frac{1}{x}, x\gt 0,$$ can I ...
4answers
348 views

0answers
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2answers
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### Evaluate $\int\limits_0^{\infty}\frac{\sin x}{x}\, dx$ [duplicate]

How can I evaluate the following improper integral: $$\int\limits_0^{\infty}\frac{\sin x}{x}\, dx$$ I have tried to evaluate this by integration by parts but failed. Please help
1answer
267 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 ...
1answer
324 views

### Proving $\int_\mathbb R\frac{\sin(x)}{x}dx = \pi$ using the residue theorem [duplicate]

I've been searching the web for a way to prove that $\int^{\infty}_{-\infty}{\sin(x)/x} = \pi$ with complex analysis, because I have a problem of consistency. I found two, carried in the following ...

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