557 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 $$\lim_{a\to\infty} \int_0^a e^{-xt}\, dt = \frac{1}{x}, x\gt 0,$$ ...
566 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 ...
591 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 ...
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 ...
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 ...
205 views

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: $$\displaystyle \int ... 1answer 125 views ### How can I evaluate \int_0^\infty \frac{\sin x}{x} \,dx? [may be duplicated] [duplicate] How can I evaluate \displaystyle\int_0^\infty \frac{\sin x}{x} \, dx? (Let \displaystyle \frac{\sin0}{0}=1.) I proved that this integral exists by Cauchy's sequence. However I can't evaluate ... 0answers 32 views ### Show that lim_{R\rightarrow \infty}\int^R_0 \frac{\sin x}{x} dx= \frac{\pi}{2}. [duplicate] I came across this qualifying exam problem and wasn't sure what to do. Using techniques of real analysis (as opposed to complex analysis) show that lim_{R\rightarrow \infty}\int^R_0 \frac{\sin x}{x} ... 6answers 2k views ### How to prove this identity \pi=\sum\limits_{k=-\infty}^{\infty}\left(\frac{\sin(k)}{k}\right)^{2}\;? How to prove this identity?$$\pi=\sum_{k=-\infty}^{\infty}\left(\dfrac{\sin(k)}{k}\right)^{2}\;$$I found the above interesting identity in the book \bf \pi Unleashed. Does anyone knows how to ... 9answers 5k views ### Proof of \int_0^\infty \left(\frac{\sin x}{x}\right)^2 dx=\frac{\pi}{2}. I am looking for a short proof that$$\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$$is also ... 6answers 5k views ### Does  \int_0^{\infty}\frac{\sin x}{x}dx  have an improper Riemann integral or a Lebesgue integral? In this wikipedia article for improper integral,$$ \int_0^{\infty}\frac{\sin x}{x}dx $$is given as an example for the integrals that have an improper Riemann integral but do not have a (proper) ... 5answers 496 views ### show that \int_{0}^{\infty} \frac {\sin^3(x)}{x^3}dx=\frac{3\pi}{8} show that$$\int_{0}^{\infty} \frac {\sin^3(x)}{x^3}dx=\frac{3\pi}{8}$$using different ways thanks for all 2answers 360 views ### Evaluating \int_0^{\infty} \text{sinc}^m(x) dx How do I evaluate$$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 ...
How do I integrate $\sin(x)/x$? I tried using integration by parts, but it led me to nowhere. Please help.
### Improper integral involving $\frac{\sin{nt}}{t}$
Assume $f(t) \colon [0,1] \to \mathbb{R}$ is a smooth function with $f(0) = 1$. Find the value of $\int_0^1\frac{\sin{nt}}{t}f(t)\,\mathrm{d}t$ as $n$ approaches infinity. I've tried approaching this ...