Linked Questions

3
votes
1answer
88 views

Proving $\int\limits _0^\infty\frac {\sin^{2n+1}x}{x}dx=\frac 1 {4^n}\binom{2n}{n}\int_0^\infty \frac {\sin{x}}{x}dx$

The question is Prove that for any $n\in\mathbb N$, $\int\limits _0^\infty\frac {\sin^{2n+1}x}{x}dx=\frac 1 {4^n}\binom{2n}{n}\int_0^\infty \frac {\sin{x}}{x}dx$ I don't have any ideas how to ...
1
vote
2answers
67 views

Parameterizing divots in contour integrals

I have recently attempted to pick up complex analysis and have been stuck on this problem for a few days: $$\int_{-\infty}^\infty \frac{\sin^2x}{x^2}\mathrm{d}x$$ Fortunately, it would seem that ...
8
votes
0answers
102 views

integrate $F(x)$: NO complex analysis, NO multivariable calculus

Suppose I have an elementary function $F(x)$ for which $\int_{-\infty}^\infty F(x) \, \text{d}x $ has an elementary value. Here 'elementary value' means anything generated by $0,1,+,-,\div,\times,\exp,...
3
votes
1answer
192 views

Help finishing this exercise!

Given the following functions: $$ F(t)= \int_0^\infty e^{-tx}\dfrac{\sin{x}}{x}\,dx, \quad t>0$$ $$ F_s(t)= \int_0^s e^{-tx}\dfrac{\sin{x}}{x}\,dx, \quad t \geq 0, s>0$$ Show that $F$ is well-...
0
votes
2answers
69 views

Evalutaion $\int _{-\infty }^{\infty }\frac{\sin\left(wz\right)}{w}\:dw$ [duplicate]

Let $$f(z)=\int _{-\infty }^{\infty }\frac{\sin\left(wz\right)}{w}\:dw$$ $$f'(z)=\int _{-\infty }^{\infty }\sin\left(wz\right)\:dw=\lim _{x\to \infty }\left(\frac{\cos\left(wz\right)}{z}\right)-\lim ...
1
vote
1answer
78 views

Using the Fourier transform to prove that $\int_0^\infty\frac{\sin u}{u}du$ is $\frac{\pi}{2}$

Prove using the Fourier integral of a gate function $f(x) = \left\{ \begin{array}{ll} 0 & T <|x|\lt \infty \ \\ \frac{1}{2} & |x| =T \\ 1 & -T < x < T \\ \...
1
vote
1answer
60 views

Is this an invalid way to compute $\int_{2\pi}^\infty \frac{\sin(x)}{x} \, dx$?

So by using tabular integration: $$\int_{2\pi}^\infty \frac{\sin(x)}{x} \, dx$$ $$= \left. -\frac{1}{x}\cos(x) - \frac{1}{x^2}\sin(x) + \frac{2}{x^3}\cos(x) + \frac{2\cdot3}{x^4}\sin(x) - \frac{2\...
1
vote
2answers
59 views

Does the improper integral $\int_0^1 \frac{1}{x}\sin\frac{1}{x^2}$ exist?

Does the improper integral $\displaystyle\int_0^1 \frac{1}{x}\sin\frac{1}{x^2}$ exist? I don't know how to use the comparison test, and I cannot find a proper comparison function.
1
vote
1answer
70 views

How to solve an indefinite integral using the Taylor series?

I am trying to show that the following integral is convergent but not absolutely. $$\int_0^\infty\frac{\sin x}{x}dx.$$ My attempt: I first obtained the taylor series of $\int_0^x\frac{sin x}{x}...
0
votes
1answer
88 views

$x\rightarrow \int_{0}^{x} \frac{\operatorname{sin}(t)}{t}$ is a bounded function

I've already proved that the improper integral $\int_{0}^{\infty}\frac{\operatorname{sin}(t)}{t}$ is convergent. I don't know its limit though... I'm asked to prove that $\begin{array}{ccccc} f ...
0
votes
1answer
38 views

convergence dominate of Lebesgue-Application

my question is: let $f(x)= \dfrac{\sin (nx)}{x}.\varphi(x)$, where $\varphi \in C^\infty_c(\mathbb{R})$. We can found an function $g \in L^1(\mathbb{R})$ such as $|f(x)| \leq g$? Please. I think that, ...
0
votes
2answers
49 views

How to study the convergence of $\int^{\infty}_{0}\frac{2\cos(t)\sin(t)}{t}dt$? [duplicate]

By using Taylor series, I managed to see that the $t \rightarrow 0$, $\frac{2\cos(t)\sin(t)}{t} = 1 -\frac{4t^2}{6}$ whose integral converges. As for the case when $t \rightarrow \infty$, I don't know ...

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