Linked Questions

9
votes
5answers
72k views

how to calculate the integral of $\sin^2(x)/x^2$ [duplicate]

Possible Duplicate: Proof for an integral involving sinc function How do I show that $\int_{-\infty}^\infty \frac{ \sin x \sin nx}{x^2} \ dx = \pi$? $\int_{-\infty}^{\infty}\sin^2(x)/x^2=\pi$ ...
9
votes
4answers
9k views

Integral of Sinc Function Squared Over The Real Line [duplicate]

I am trying to evaluate $$\int_{-\infty}^{\infty} \frac{\sin(x)^2}{x^2} dx $$ Would a contour work? I have tried using a contour but had no success. Thanks. Edit: About 5 minutes after posting this ...
3
votes
1answer
2k views

Improper integral of $\sin t/t$ [duplicate]

Possible Duplicate: Proof for an integral involving sinc function Prove that: $\displaystyle \int_{0}^{\infty }\frac{\sin t}{t}dt=\int_{0}^{\infty }\frac{\sin^{2}t}{t^{2}}dt$ Thank you in ...
5
votes
3answers
4k views

Calculating $\int_0^\infty \frac {\sin^2x}{x^2}dx$ using the Residue Theorem. [duplicate]

I am trying to compute the following integral using the Residue Theorem but am quite stuck: $$\int_0^\infty \frac{\sin^2x}{x^2}dx$$ I have tried applying Jordan's lemma, having written $\sin(x)$ as $\...
5
votes
4answers
456 views

Integrating $ \int \limits_{-\infty}^{\infty} \frac{\sin^2(x)}{x^2} \operatorname d\!x $ [duplicate]

I'm trying to evaluate $\displaystyle \int \limits_{-\infty}^{\infty} \dfrac{\sin^2(x)}{x^2} \operatorname d\!x $. My first though was to use residue calculus, since we've got the pole of order 2 ...
6
votes
2answers
631 views

Integrate: $\int_0^\infty \frac{\sin^2(x)}{x^2}dx$ [duplicate]

I am trying to integrate $\displaystyle \int_0^\infty \frac{\sin^2(x)}{x^2} dx$ by method of contour. I am considering the following contour but I am not being able to. Also I am not sure if it's ...
3
votes
2answers
430 views

How the calculate $\int_0^{+\infty} \frac{\sin^2 x}{x^2} \,\mathrm{d} x$? [duplicate]

Just as the title say, consider the integral: $$I=\int_0^{+\infty} \frac{\sin^2 x}{x^2} \,\mathrm{d} x=\frac{1}{2}\int_{-\infty}^{+\infty} \frac{\sin^2 x}{x^2} \,\mathrm{d} x,$$ how to apply the ...
2
votes
1answer
441 views

Integration of $\int_0^\infty\frac{\sin^2x}{x^2}dx$ [duplicate]

I have been trying very hard to find the answer to the following integral$$\int_0^\infty\frac{\sin^2x}{x^2}dx$$ given that $$\int_0^\infty\frac{\sin x\cos x}{x}dx = \frac{\pi}{4}$$
6
votes
0answers
224 views

Questions regarding $\int_0^\infty\frac{\sin^2x}{x^2}dx$ [duplicate]

question: $\int_0^\infty\frac{\sin^2x}{x^2}dx$ is equal to (A)$\int_0^\infty\frac{\sin x}{x}dx$ (B)$\int_0^\infty\frac{\cos x}{x}dx$ (C)$\int_0^\infty\frac{\cos^2x}{x^2}dx$ (D)...
1
vote
0answers
212 views

How to integrate $\frac{\sin^2 x}{x^2}?$ [duplicate]

Possible Duplicate: Proof for an integral involving sinc function How can one calculate $\displaystyle\int_0^{\infty} \frac{\sin^2 x}{x^2} dx$? I know that the answer is $\pi/2$ and I imagine we ...
1
vote
2answers
98 views

Prove that $ \int_0^\infty (\frac{\sin x}{x})^2 = \frac{\pi}{2}$. [duplicate]

I need to prove that $ \int_0^\infty (\frac{\sin x}{x})^2 = \frac{\pi}{2}$. I have proved that $\sum_1^\infty \frac {\sin^2(n \delta)}{n^2 \delta}=\frac{\pi-\delta}{2}$ for $0<\delta<\pi$ and ...
0
votes
1answer
152 views

Evaluate $\int_{-\infty }^{\infty } \frac{\sin^2x}{x^2} \, dx$ [duplicate]

How to evaluate: $$\int_{-\infty }^{\infty } \frac{\sin^2x}{x^2} \, dx$$ This is a problem in my complex variables course. This integral is intended to be solved by contour, but I’m struggling ...
1
vote
1answer
137 views

Integral $\int_0^{\infty} \left( \frac{\sin x}{x} \right)^2dx$. [duplicate]

Consider the contour $C_{\epsilon,R} = \Gamma_R \cup \gamma_{\epsilon} \cup [-R,-\epsilon] \cup [\epsilon, R]$ and $f(z)=\frac{1-e^{2iz}}{z^2}$. By Cauchy's Theorem we have $$\int_{C_{\epsilon,R}}f(z)...
3
votes
1answer
62 views

How to calculate the integral [duplicate]

$$\int_{0}^{+ \infty } \frac{\sin^{2} x}{x^{2}}dx$$ I suppose we should start from the known one:$$\int_{0}^{+ \infty } \frac{\sin x}{x}dx = \pi /2$$
208
votes
28answers
95k views

Evaluating the integral $\int_0^\infty \frac{\sin x} x \,\mathrm dx = \frac \pi 2$?

A famous exercise which one encounters while doing Complex Analysis (Residue theory) is to prove that the given integral: $$\int_0^\infty \frac{\sin x} x \,\mathrm dx = \frac \pi 2$$ Well, can ...

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