# Linked Questions

5answers
3k views

### Are there any ways to evaluate $\int^\infty_0\frac{\sin x}{x}dx$ without using double integral? [duplicate]

Are there any ways to evaluate $\int^\infty_0\frac{\sin x}{x}dx$ without using double integral? I can't find any this kind of solution. Can anyone please help me? Thank you.
3answers
1k views

### Evaluating $\lim_{b\to\infty} \int_0^b \frac{\sin x}{x}\, dx= \frac{\pi}{2}$ [duplicate]

Possible Duplicate: Solving the integral $\int_{0}^{\infty} \frac{\sin{x}}{x} \ dx = \frac{\pi}{2}$? Using the identity $$\lim_{a\to\infty} \int_0^a e^{-xt}\, dt = \frac{1}{x}, x\gt 0,$$ can I ...
4answers
348 views

0answers
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2answers
242 views

### Evaluate $\int\limits_0^{\infty}\frac{\sin x}{x}\, dx$ [duplicate]

How can I evaluate the following improper integral: $$\int\limits_0^{\infty}\frac{\sin x}{x}\, dx$$ I have tried to evaluate this by integration by parts but failed. Please help
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 ...
1answer
324 views

### Proving $\int_\mathbb R\frac{\sin(x)}{x}dx = \pi$ using the residue theorem [duplicate]

I've been searching the web for a way to prove that $\int^{\infty}_{-\infty}{\sin(x)/x} = \pi$ with complex analysis, because I have a problem of consistency. I found two, carried in the following ...
1answer
313 views

### Rigorously proving $\displaystyle\int^{\infty}_0 \frac{\sin{x}}{x} dx= \frac{\pi}{2}$ [duplicate]

I want to prove the famous formula: $\displaystyle\int^{\infty}_0 \frac{\sin{x}}{x} dx = \frac{\pi}{2}.$ There are many ways to do it, for example, by some Fourier analysis. But how about a simple ...
1answer
223 views

### how to evaluate this definite integral $\int_0^\infty\frac{\sin^2(x)}{x^2}dx$? [duplicate]

For $\int_{0}^{\infty}\frac{\sin^2(x)}{x^2}dx$. I considered using residue theorem. But since the function inside is holomorphic except for a removable singularity at the origin. So whatever contour I ...
2answers
119 views

### Prove the relation $\frac{1}{x}$=$\int^\infty_0$ $e^{-xt}$ dt, for $x>0$. Use it to prove $\int^\infty_0$ $\frac{\sin(x)}{x}$ dx = $\frac{\pi}{2}$ [duplicate]

Prove the relation $$\frac{1}{x} = \int^\infty_0 e^{-xt}\, \text{d}t, \text{ for } x>0.$$ Use it to prove $$\int^\infty_0\frac{\sin(x)}{x}\, \text{d}x = \frac{\pi}{2}.$$ "Hint: Use ...
0answers
249 views

Is there an elementary way to proof that $$\int_0^\infty \frac{\sin{x}}{x} \mathrm{d}{x} = \frac{\pi}{2}$$ or equivalently $$\int_{-\infty}^{+\infty} \frac{\sin{x}}{x} \mathrm{d}{x} = \pi \;\;\; \... 1answer 124 views ### Alternative approach to the computation of \int_0^\infty\frac{\sin(nx)}xdx [duplicate] I am not able to obtain the value of:$$ \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.$$And by using ... 0answers 146 views ### Trigonometric Integral (Sine Integral) [duplicate] How to evaluate the following integral ?$$\boxed {\int_{0}^\infty \frac{\sin(t)}{t}\mathrm dt} $$1answer 78 views ### Are there any ways of solving this intriguing integral? [duplicate] I know a good number of methods to solve this integral like Feynman Trick, Laplace transform etc.$$\int_0^{\infty} \dfrac{\sin x}{x} \mathrm{d}x = \dfrac{π}{2} $$Do y'all know some creative and ... 0answers 74 views ### Integration of \sin x/x [duplicate] How to evaluate \displaystyle \int_0^\infty \frac{\sin x} x \, dx I have tried substitutions and integration by parts, but unable to integrate it. 0answers 65 views ### Compute \int_0^\infty \frac{\sin(x)}{x}dx without residue theorem. [duplicate] Is there a way to compute$$\int_0^\infty \frac{\sin(x)}{x}dx$$without residue theorem ? 0answers 54 views ### Need some help with an integral [duplicate] I was reading on integration techniques and a came across an example that is shown quite a lot around the web.$$ \int_0^\infty \frac {\sin(x)}{x}\,dx $$I do know the answer must be \frac {\pi}{... 0answers 42 views ### Calculate \int _{x=0}^{\infty} \frac{\sin(x)}{x} with the function \frac{e^{iz}}{z} [duplicate] I want to calculate \int _{x=0}^{\infty} \frac{\sin(x)}{x} with the function f(z) = \frac{e^{iz}}{z}. I thought about using the closed path \Gamma = \gamma _1 + \gamma _R + \gamma _2 + \gamma _{\... 1answer 41 views ### Evaluate \int_{0}^{\infty} \frac{\sin{x}}{x}\,dx [duplicate] how do I solve this integral ?$$\int_{0}^{\infty} \frac{\sin{x}}{x}\,dx$$how to start ? 0answers 41 views ### How to prove \int_0^{\infty} \frac{\sin x}{x} \, dx = \frac{\pi}{2} [duplicate] How to prove \displaystyle \int_0^{\infty} \frac{\sin x}{x} \, dx = \frac{\pi}{2}? How do I proceed? 0answers 39 views ### Help needed in solving this problem of Fourier Transform [duplicate] I want to know why$$\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}$$where f(x) is \begin{cases} ... 14answers 16k views ### Proof of \int_0^\infty \left(\frac{\sin x}{x}\right)^2 \mathrm dx=\frac{\pi}{2}. I am looking for a short proof that$$\int_0^\infty \left(\frac{\sin x}{x}\right)^2 \mathrm dx=\frac{\pi}{2}.$$What do you think? It is kind of amazing that$$\int_0^\infty \frac{\sin x}{x} \mathrm ...

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