Linked Questions

0
votes
1answer
4k views

Dirichlet integral. [duplicate]

I want to prove $\displaystyle\int_0^{\infty} \frac{\sin x}x \,\mathrm{d}x = \frac \pi 2$, and $\displaystyle\int_0^{\infty} \frac{|\sin x|}x \,\mathrm{d}x \to \infty$. And I found in wikipedia, but ...
0
votes
1answer
267 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 ...
-1
votes
1answer
124 views

Alternative approach to the computation of $\int_0^\infty\frac{\sin(nx)}xdx$ [duplicate]

I am not able to obtain the value of:$$ \int_{0}^{\infty}\dfrac{\sin(nx)}{x}dx ,$$ By changing the order of integration of: $$\int_{0}^{\infty}\int_{0}^{\infty}e^{-xy}\sin(nx) dx dy.$$ And by using ...
-1
votes
1answer
686 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 a ...
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
0answers
147 views

Trigonometric Integral (Sine Integral) [duplicate]

How to evaluate the following integral ? $$\boxed {\int_{0}^\infty \frac{\sin(t)}{t}\mathrm dt} $$
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 _{\...
1
vote
0answers
74 views

Integration of $\sin x/x$ [duplicate]

How to evaluate $\displaystyle \int_0^\infty \frac{\sin x} x \, dx$ I have tried substitutions and integration by parts, but unable to integrate it.
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^+}\...
1
vote
0answers
39 views

Help needed in solving this problem of Fourier Transform [duplicate]

I want to know why $$\int_0^\infty \frac{\cos wx\sin w}{w}\,dw = \frac{\pi}{4}?$$ at x=0,given that $$\int_0^\infty \frac{\cos wx\sin w}{w}\,dw = \frac{\pi f(x)} {2}$$ where f(x) is \begin{cases} ...
1
vote
0answers
250 views

Are there elementary ways to proof $\int_0^\infty \frac{\sin{x}}{x} \mathrm{d}{x}=\frac{\pi}{2}$? [duplicate]

Is there an elementary way to proof that $$\int_0^\infty \frac{\sin{x}}{x} \mathrm{d}{x} = \frac{\pi}{2}$$ or equivalently $$\int_{-\infty}^{+\infty} \frac{\sin{x}}{x} \mathrm{d}{x} = \pi \;\;\; \...
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}{...
0
votes
0answers
937 views

Improper integration of sin(z)/z [duplicate]

I need to calculate this $\int_{-\infty}^\infty \frac{sin(z)}{z}dz$ using the residue of $\frac{sin(z)}{z}$. Then I write $\int_{-\infty}^\infty \frac{sin(z)}{z}dz$ = $\lim_{R\to\infty}Im(\int_{-R}^R ...
0
votes
0answers
41 views

How to prove $\int_0^{\infty} \frac{\sin x}{x} \, dx = \frac{\pi}{2}$ [duplicate]

How to prove $\displaystyle \int_0^{\infty} \frac{\sin x}{x} \, dx = \frac{\pi}{2}$? How do I proceed?
0
votes
0answers
65 views

Compute $\int_0^\infty \frac{\sin(x)}{x}dx$ without residue theorem. [duplicate]

Is there a way to compute $$\int_0^\infty \frac{\sin(x)}{x}dx$$ without residue theorem ?

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