12 questions linked to/from Create a Huge Problem
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### 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 ...
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### 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 ...
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 $\... 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 $$\int_0^\infty e^{-\alpha x^2}dx = \... 6answers 640 views ### Evaluating the integral \int_0^\infty \frac{x \sin rx }{a^2+x^2} dx using only real analysis Calculate the integral$$ \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}. $$Edit: I was able to solve the integral ... 4answers 332 views ### How do I evaluate the integral \int_0^{\infty}\frac{x^5\sin(x)}{(1+x^2)^3}dx? I have no idea how to start, it looks like integration by parts won't work.$$\int_0^{\infty}\frac{x^5\sin(x)}{(1+x^2)^3}dx$$If someone could shed some light on this I'd be very thankful. 3answers 218 views ### Value of the integral \int_{\mathbb{R}} \frac{x\sin {(\pi x)}}{(1+x^2)^2} How do we evaluate the integral$$I=\displaystyle\int_{\mathbb{R}} \dfrac{x\sin {(\pi x)}}{(1+x^2)^2}$$I have wasted so much time on this integral, tried many substitutions (x^2=t, \ \pi x^2=t). ... 2answers 182 views ### How to integrate \int_{-\infty} ^\infty \frac{\cos(xy)}{x^2+1}dx Is there a standard trick to compute this integral for y\ge 0? \int_{-\infty} ^\infty \frac{\cos(xy)}{x^2+1}dx = \int_{-\infty}^{\infty}\frac{y \cos(x)}{x^2+y^2} Hopefully the same trick could ... 2answers 332 views ### Integral without residues How do I do this integral without using complex variable theorems? (i.e. residues)$$\lim_{n\to \infty} \int_0^{\infty} \frac{\cos(nx)}{1+x^2} \, dx$$2answers 534 views ### Computing the inverse Fourier transform of \frac{1}{1+|\xi|^2} for \xi \in \mathbb{R}^n. I'm trying to compute the integral$$ \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 + |\... 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 ...
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?