Linked Questions

1
vote
1answer
32 views

How to know whether to choose the x-bound or the y-bound for this triple integral

In my textbook for calculus 3, I have been working on example of the triple integral. Though I do know polar, cylindrical, spherical coordinates, this section of the book expects you to work with ...
13
votes
2answers
13k views

Cauchy distribution characteristic function

I know that it's easy to calculate integral $\displaystyle\int_{-\infty}^\infty \frac{e^{itx}}{\pi(1+x^2)} \, dx$ using residue theorem. Is there any other way to calculate this integral (for someone ...
1
vote
1answer
89 views

Can $\int_{0}^{\infty}\frac {\cos{x}}{(1 + x^2)} dx$ be evaluated without complex analysis?

Can the integral $\int_{0}^{\infty}\frac {\cos{x}}{(1 + x^2)} dx$ be evaluated by differentiation under integral or any other method without involving complex analysis? I tried using the function $\...
5
votes
3answers
176 views

Evaluate Gauss-like Integral

Evaluate Integral $$\int_0^\infty e^{-ay^{2}-\frac{b}{y^2}}dy $$ Where a and b are real and positive. This integral is eerily similar to the Gaussian integral $$\int_0^\infty e^{-\alpha x^2}dx = \...
5
votes
6answers
640 views

Evaluating the integral $\int_0^\infty \frac{x \sin rx }{a^2+x^2} dx$ using only real analysis

Calculate the integral$$ \int_0^\infty \frac{x \sin rx }{a^2+x^2} dx=\frac{1}{2}\int_{-\infty}^\infty \frac{x \sin rx }{a^2+x^2} dx,\quad a,r \in \mathbb{R}. $$ Edit: I was able to solve the integral ...
6
votes
4answers
332 views

How do I evaluate the integral $\int_0^{\infty}\frac{x^5\sin(x)}{(1+x^2)^3}dx$?

I have no idea how to start, it looks like integration by parts won't work. $$\int_0^{\infty}\frac{x^5\sin(x)}{(1+x^2)^3}dx$$ If someone could shed some light on this I'd be very thankful.
1
vote
3answers
218 views

Value of the integral $\int_{\mathbb{R}} \frac{x\sin {(\pi x)}}{(1+x^2)^2}$

How do we evaluate the integral $$I=\displaystyle\int_{\mathbb{R}} \dfrac{x\sin {(\pi x)}}{(1+x^2)^2}$$ I have wasted so much time on this integral, tried many substitutions $(x^2=t, \ \pi x^2=t)$. ...
5
votes
2answers
182 views

How to integrate $\int_{-\infty} ^\infty \frac{\cos(xy)}{x^2+1}dx$

Is there a standard trick to compute this integral for $y\ge 0$? $\int_{-\infty} ^\infty \frac{\cos(xy)}{x^2+1}dx = \int_{-\infty}^{\infty}\frac{y \cos(x)}{x^2+y^2}$ Hopefully the same trick could ...
4
votes
2answers
332 views

Integral without residues

How do I do this integral without using complex variable theorems? (i.e. residues) $$\lim_{n\to \infty} \int_0^{\infty} \frac{\cos(nx)}{1+x^2} \, dx$$
4
votes
2answers
534 views

Computing the inverse Fourier transform of $\frac{1}{1+|\xi|^2}$ for $\xi \in \mathbb{R}^n$.

I'm trying to compute the integral $$ \int_{\large\mathbb{R}^n} \frac{ e^{\large ix \cdot \xi}}{1 + |\xi|^2} ~d^n\xi. $$ I know that for an integral like $$\int_{\large\mathbb{R}^n} \frac{ 1}{1 + |\...
3
votes
2answers
90 views

I want to prove $k(x,t)=\frac{1}{\sqrt{4\pi t} } e^{\frac{-x^2}{4t}} $

I have this integral $$u(x,t)=\int _{-\infty}^{\infty} f(\eta)\left[\frac{1}{2\pi}\int _{-\infty}^{\infty}e^{iw(x-\eta)-w^2t}\ dw\right]\ d\eta=\int _{-\infty}^{\infty}k(x-\eta,t)f(\eta)\ d\eta$$ I ...
1
vote
2answers
120 views

How to integrate this fourier transform?

I want to integrate $$\int_{-\infty}^{\infty} \frac{e^{itx}}{{1+x^2}} dx.$$ I don't see how substitution or integration by parts could help here. Does anybody know how to do this?