Linked Questions

3
votes
2answers
227 views

$\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 ...
2
votes
2answers
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 ...
0
votes
3answers
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 ...
1
vote
2answers
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. ...
-1
votes
3answers
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 $$ \int_{-\infty}^\infty 2\,\frac{\cos(w)\sin(w)}{w}\ dw ? $$
5
votes
2answers
240 views

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 ...
2
votes
2answers
128 views

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 ...
1
vote
1answer
209 views

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.
0
votes
3answers
124 views

$\lim_{n\rightarrow\infty} \int_n^{n+1}\frac{\sin x}x \, dx$

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. ...
1
vote
1answer
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 ...
2
votes
2answers
73 views

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 $\int \frac{\sin(\pi x)}{x}=\frac{1-\cos(\pi x)}{\pi x}+ \int \frac{1-\cos(\pi x)}{\pi x^2} dx$.
1
vote
2answers
104 views

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.
0
votes
1answer
80 views

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.
1
vote
0answers
161 views

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^+}\...
2
votes
1answer
159 views

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 ...

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