Linked Questions

3
votes
6answers
165 views

Provide a different method of proving:$\int_{-\infty}^{\infty}{1\over [\pi(x+e^{\pi})^2+\pi^{1/3}]^2}dx={1\over 2}$

Accidentally founded this particular integral producing a rational number I can't be for sure it is correct, so can one provide a proof of it. $$\int_{-\infty}^{\infty}{1\over [\pi(x+e^{\pi})^2+\pi^...
4
votes
3answers
645 views

Find the principal value of $\int_{-\infty}^{\infty}\frac{1-\cos x}{x^2}\,\mathrm dx$

How to find the Cauchy principal value of the following integral $$\int_{-\infty}^{\infty}\frac{1-\cos x}{x^2}\,\mathrm dx$$ How to start this problem?
4
votes
4answers
284 views

Mistake with using residue theory for calculating $\int_{-\infty}^{\infty}\frac{\sin(x)}{x}dx$

In physics, we do contour integrals in a not so rigorous way most of the times. Now, I want to use a trick to compute $$\int_{-\infty}^{\infty}\frac{\sin(x)}{x}dx$$ using residue theory. I find: $$\...
5
votes
2answers
299 views

Two sums for $\pi$

I was looking for a simple way to evaluate the integral $\int_0^\infty \frac{\sin x}{x}dx$ ( a belated look at this question). There are symmetries to be exploited, for one thing. So I had an idea, ...
6
votes
2answers
314 views

Difficulties understanding a proof of $\int_0^{\infty} \frac{\sin(x)}{x} \, dx = \frac{\pi}{2}$

I got a homework and I've trying to do this problem about 2 days, but I "lost my fight". So I turn to you. I have to prove that $$\int _0^\infty \frac{\sin (x)}{x} \, dx = \frac{\pi}{2}.$$ I can't use ...
2
votes
5answers
643 views

Showing that $\int_0^{+\infty} \frac{\sin(1/x)}{x}\ dx$ converges

How can I show that $\int_0^\infty \frac{\sin(1/x)}{x}\ dx$ converges? I have that $\sin(x)\leq x$ for $x\geq 0$ so then $\sin(1/x)\leq 1/x$ for $x\geq 0$. It follows then that $\int_1^\infty \frac{\...
8
votes
3answers
135 views

How to bound $\int_{0}^{a}{\frac{1-\cos x}{x^2}}$?

I was trying to prove $$\left|\int_{0}^{a}{\frac{1-\cos{x}}{x^2}}dx-\frac{\pi}{2}\right|\leq \frac{3}{a}$$ or $\leq \frac{2}{a}$. My work: I would like to use Fubini's theorem to prove it. I ...
2
votes
2answers
319 views

Prove that: $ \int_{0}^{\infty} \frac{2 x \sin x+\cos 2x-1}{2 x^2} = 0$

How would you prove that? $$ \int_{0}^{\infty} \frac{2 x \sin x+\cos 2x-1}{2 x^2} dx= 0$$ I'm looking for a solution at high school level if possible. Thanks.
6
votes
2answers
637 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 ...
5
votes
3answers
231 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 ...
3
votes
2answers
432 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 ...
5
votes
2answers
534 views

uniform bound for sine integral function

Prove that for any $0<a<b$, $$ \left|\int_a^b\frac{\sin x}{x}\,dx\right|\le4 $$ Here is my approach. I used integration by parts to prove that LHS is bounded by $3$ when $a\ge 1$. I will be done ...
3
votes
3answers
213 views

Prove that $\int\limits_1^{\infty} \frac{\cos(x)}{x} \, \mathrm{d}x$ converges

Prove the convergence of $$\int\limits_1^{\infty} \frac{\cos(x)}{x} \, \mathrm{d}x$$ First I thought the integral does not converge because $$\int\limits_1^{\infty} -\frac{1}{x} \,\mathrm{d}x \...
4
votes
1answer
205 views

How may be prove that $\int_{-\infty}^{+\infty}\sin(\cosh x)\cos(\sinh x)\mathrm dx={\pi\over 2}?$

Given this integral $$\int_{-\infty}^{+\infty}\sin(\cosh x)\cos(\sinh x)\mathrm dx={\pi\over 2}\tag1$$ My try: Recall $$2\sin(A)\cos(B)=\sin(A-B)+\sin(A+B)$$ $(1)$ becomes $$\frac12\int_{-\...
10
votes
1answer
265 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 $(...

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