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### $\int_{0}^{\infty}\frac{\sin x}{x}dx$ converges

I want to show that $\int_{0}^{\infty}\frac{\sin x}{x}dx$ converges. I am facing difficluty in my last step Although there are many proofs regarding this but doubt is hasnt been addressed. My ...
195 views

### $\lim_{A \to \infty} \int_0^{A} \int_0^{\infty} \sin(x) e^{-xt}dtdx$

I would like to compute the following integral: $$\lim_{A \to \infty} \int_0^{A} \int_0^{\infty} \sin(x) e^{-xt}dtdx \qquad (1)$$ I would like to swap the order of integration because then the ...
149 views

### Find whether $\int_{1}^{\infty} \frac{sin(x)}{x} dx$ is converging or not

Find whether $\int_{1}^{\infty} \frac{sin(x)}{x} dx$ is converging or not. I tried to use comparison test or limit comparison test but could't find a suitable function. How can I determine what type ...
145 views

### Will integral be $\frac{\pi}{2}$?

Show that $\int \frac{1-cosx}{x^2}\ dx=\frac{\pi}{2}$. I used Taylor's series for cosx to find integral but I don't see intergal becoming equal to $\frac{\pi}{2}$ without any limits of integration. ...
191 views

### Evaluation of $\int_{-\infty}^\infty \frac{\sin(at)}t\ dt$

I just need to know if this integral converges. If it does, what is the value? If not can anyone tell me what is the value of the $$\int_{-\infty}^\infty 2\,\frac{\cos(w)\sin(w)}{w}\ dw ?$$
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### Can one solve $\int_{0}^{\infty} \frac{\sin(x)}{x} dx$ *from its Taylor series antiderivative directly*?

This question was inspired by this question: Evaluating the integral $\int_{0}^{\infty} \frac{\sin{x}}{x} \ dx = \frac{\pi}{2}$? Well, can anyone prove this without using Residue theory. I ...
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### Evaluating the integral: $\int\limits_{0}^{\infty}\left(\frac{\sin(ax)}{x}\right)^2 dx , a \neq 0$

Integrate from 0 to infinity $$\int\limits_{0}^{\infty}\left(\frac{\sin(ax)}{x}\right)^2 dx , a \neq 0$$ I tried evaluating the indefinite integral of that function using Sine integral. But I ...
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### Proof that $\int_0^\infty x^{d-4}\sin x\, dx = \cos \frac{\pi d}{2} \Gamma(d-3)$, for $2<\Re(d)<4$?

Can one prove that $$\int_0^\infty x^{d-4}\sin x\, dx = \cos \frac{\pi d}{2} \Gamma(d-3),\text{ for }2<\Re(d)<4?$$ I would prefer using the methods of contour integration.
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### $\lim_{n\rightarrow\infty} \int_n^{n+1}\frac{\sin x}x \, dx$

I am asked to calculate $\displaystyle \lim_{n\rightarrow\infty} \int_n^{n+1}\frac{\sin x} x \, dx$. Before letting the $\lim$ to confuse me I used integration by parts, but it didn't get me far. ...
156 views

### Why can't the indefinite integral $\int\frac{\sin(x)}{x}\mathrm dx$ be found? [duplicate]

I came across a list of functions in my calculus textbook whose indefinite integral cannot be found. It was written that the integral $$\int \frac{\sin(x)}{x} dx$$ cannot be evaluated without any ...
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### Convergence of $\int_{0}^{+\infty} \frac{\sin(\pi x)}{x} dx$

I want to prove that $$\int_{0}^{+\infty} \frac{\sin(\pi x)}{x} dx$$ is convergent; I know that $\int \frac{\sin(\pi x)}{x}=\frac{1-\cos(\pi x)}{\pi x}+ \int \frac{1-\cos(\pi x)}{\pi x^2} dx$.
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### How can I compute $\int_0^\infty {\sin x \over x} dx$ from computing $\int_0^\infty e^{-xt} {{\sin x} \over x} dx$

This is what I tried. I let $\int_0^\infty e^{-xt} {\sin x \over x} dx= F(t)$ and computed $f(t) = \int_0^\infty (-x) * e^{-xt} * (sinx/x) dx$ but I couldn't get anything more. Please help me.
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### Prove $\int_{0}^{\infty} \frac{\sin{x} dx}{x}$ converges.

The fact that $\int_{0}^{\infty} \frac{\sin{x} dx}{x}$ converges has been proven multiple times for example: here However I am wondering what is the way of proving it using Cauchy-Bolzano criterion.
### Evaluate $\int_{0}^{+\infty}\frac{\sin x}{x}dx$ [duplicate]
$$\int_{0}^{+\infty}\frac{\sin x}{x}dx$$ My start: Zero is a singular point, Let's define $g(z):=(e^{iz})\big/z$ \int_{\Gamma}\frac{e^{iz}}{z}dz=\\ \int_{-R}^{-r}\frac{e^{ix}}{x}dx+\int_{C_r^+}\...
### How to derive an explicit formula for $\sum \frac{e^{i n \theta}}{n}?$
Suppose $\theta$ is not an integer multiple of $\pi$. The series $\left | \sum e^{i n \theta} \right |$ is bounded above by $\frac{1}{|\sin \theta|}$ and, as $\left ( \frac{1}{n} \right )$ is ...