Linked Questions

1
vote
2answers
58 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
41 views

Evaluate $\int_{0}^{\infty} \frac{\sin{x}}{x}\,dx$ [duplicate]

how do I solve this integral ? $$\int_{0}^{\infty} \frac{\sin{x}}{x}\,dx$$ how to start ?
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}...
1
vote
0answers
42 views

Calculate $\int _{x=0}^{\infty} \frac{\sin(x)}{x}$ with the function $\frac{e^{iz}}{z}$ [duplicate]

I want to calculate $\int _{x=0}^{\infty} \frac{\sin(x)}{x}$ with the function $f(z) = \frac{e^{iz}}{z}$. I thought about using the closed path $\Gamma = \gamma _1 + \gamma _R + \gamma _2 + \gamma _{\...
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,...
5
votes
3answers
235 views

Using Laplace Transforms to solve $\int_{0}^{\infty}\frac{\sin(x)\sin(x/3)}{x(x/3)}\:dx$

So, I've come across the following integral (and it's expansion) many times and in my study so far, Complex Residues have been used to evaluate it. I was hoping to find an alternative approach using ...
0
votes
0answers
54 views

Need some help with an integral [duplicate]

I was reading on integration techniques and a came across an example that is shown quite a lot around the web. $$ \int_0^\infty \frac {\sin(x)}{x}\,dx $$ I do know the answer must be $\frac {\pi}{...
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 ...
0
votes
1answer
78 views

Are there any ways of solving this intriguing integral? [duplicate]

I know a good number of methods to solve this integral like Feynman Trick, Laplace transform etc. $$\int_0^{\infty} \dfrac{\sin x}{x} \mathrm{d}x = \dfrac{π}{2} $$ Do y'all know some creative and ...
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 ...
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, ...
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 ...
2
votes
3answers
195 views

Confirm that $\int_{0}^{\infty}t^{-1}\sin t dt=\pi/2$ [duplicate]

Confirm that $\int_{0}^{\infty}t^{-1}\sin t dt=\pi/2$. The guide book I am using gives the following help: Consider $\int_{\gamma}z^{-1}e^{iz}dz$, where for $0<s<r<\infty$ the contour of ...
10
votes
1answer
267 views

How to show that $\int_{0}^{\infty}{\sin(x)\sin(2x)\sin(4x)\cdots\sin(2^kx)\over x^{k+1}}\mathrm dx=2^{0.5(k^2-k-2)}\pi?$

$$\int_{0}^{\infty}{\sin(x)\sin(2x)\sin(4x)\cdots\sin(2^kx)\over x^{k+1}}\mathrm dx=2^{0.5(k^2-k-2)}\pi\tag1$$ $k\ge0$ Experimental using wolfram integrator, let me to conclude the closed form for $(...
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 ...

15 30 50 per page