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I apologize in advance if this is a very soft question. I won't be surprised or offended if I can't get a good answer.

One of my favorite things to do in my spare time, when I'm feeling analytical of course, is to evaluate integrals, both definite and indefinite. However, I've had little success here on Math.SE trying to find integrals that meet my criteria.

Either the integral in question will be way beyond the methods that I understand to evaluate it (typically using contour integration), or is so mind-numbingly trivial that I can't be bothered writing it down. I've scoured the internet for some interesting integrals, and I found the MIT Integration Bee, but those aren't really that hard either. There are some decent ones in my multivariable calculus textbook, but I'm starting to run out of those too.

Is there any specific place I should be looking for interesting, tough but doable without complex analysis? Specifically ones where we can evaluate through tricks like clever substitutions or exploitation of symmetry or changing coordinates, etc.

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    $\begingroup$ I'd still say have a look at this site (the integral tags). Most of the integrals that appears and that are calculated with complex analysis tools can be calculated also with real methods. Introducing a parameter and differentiating with respect to it is a useful method. Also, please give one example of the type of integral you are looking for. Finally, if you only practice on the types of integrals you already manage, you will never beat Ron Gordon/the other gurus on this site... $\endgroup$ – mickep May 11 '15 at 15:03
  • $\begingroup$ folk.ntnu.no/oistes/Diverse/Integral/Integral%20Kokeboken.pdf $\endgroup$ – N3buchadnezzar May 11 '15 at 15:47
  • $\begingroup$ @mickep, I'll have to look harder and maybe not be so daunted by answers with complex analysis. I've been trying to find a good text for that subject, specifically so I can tackle hairy integrals, but I've had a hard time understanding the concepts before contour integration. Ron Gordon is exactly the kind of skill I aspire to, but certainly not within the next 10 year or so. His is a skill borne out of a deep appreciation for analytic problem solving, coupled with an enormous amount of experience. His answers always leave me dazzled. $\endgroup$ – Tyg13 May 11 '15 at 17:46
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    $\begingroup$ In addition to the other valid suggestions, perhaps also find some people with a similar interest (they occur here and there persistently) and create or find integrals for each other? I find coming up with challenging integrals is a nice skill to polish (a little similar to solving them), and if you find some other individuals, you could share problems with each other. $\endgroup$ – izœc May 12 '15 at 7:58
  • $\begingroup$ math.stackexchange.com/questions/765198/… Here are some answers. In the link I posted earlier are a large number of semi-hard problems, look at page 11^2 as an example. Basically being good at something boils down to doing it a lot. Once you have solved a few hundred integrals, you will be much more profficient at it. $\endgroup$ – N3buchadnezzar May 12 '15 at 15:37
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I'll give you as a challenge one of my favorite double integration problems. There are lots of ways to do this (including no calculus at all, as Archimedes would have done it); see if you can find the most elegant and efficient.

Find the volume of the region inside all three cylinders $$x^2+y^2=a^2, \quad x^2+z^2=a^2, \quad y^2+z^2=a^2.$$

P.S. For a bit of challenge in the single-variable setting, you might try the "Potpourri" problems in Spivak's Calculus (in the latter editions, it's #8 in Chapter 19).

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    $\begingroup$ Man, that is hard to visualize. "You can see new things by looking at earth under a microscope or drawing the shape of the solid made when 3 circular rods of equal thickness intersect at right angles" - Christopher, The Curious Incident of the Dog in the Night-Time $\endgroup$ – MCT May 12 '15 at 14:40
  • $\begingroup$ @Soke: I don't want to ruin anyone's fun, but I'd recommend drawing only the portion in the first octant. $\endgroup$ – Ted Shifrin May 12 '15 at 15:07
  • $\begingroup$ One could also do the suggested calculation, but for the region inside two of the three cylinders. $\endgroup$ – mickep May 12 '15 at 18:00
  • $\begingroup$ @mickep, sure, but that's an easy single integral problem. $\endgroup$ – Ted Shifrin May 12 '15 at 18:11
  • $\begingroup$ Agreed, but we are not really sure on OP's skills yet. Also, it could be good as a warm up. $\endgroup$ – mickep May 12 '15 at 18:24
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Since I'm not familiar with the content of Calculus III, this might be on the wrong level. I'm sorry if they are too simple. Then we can iterate to get to a better level...

1) Let $a>0$. Find the area of the region enclosed by the curve $x^3+y^3-3axy=0$. The figure below shows the domain in the case $a=1$.

Funny domain

2) Let $a>-1$ and $b>-1$. Calculate the integral $$ \int_0^1\frac{x^b-x^a}{\ln x}\,dx. $$

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  • $\begingroup$ I admit, I was a little more confident in my abilities than I should have been, heh. For the first problem, I tried to separate the variables, and then ,failing that, finding a parameterization in $t$ for $x$ and $y$. Both of which were unsuccessful. I also tried viewing it as a contour of $f(x,y) = x^3 + y^3 - 3axy$ but couldn't see how that would help evaluate the integral. For the 2nd problem, I got it in the form $\int_{-\infty}^{0} \dfrac{e^{(a+1)u} - e^{(b+1)u}}{u}\;\mathrm{d}u, a\neq b$, but as far as I know, this is just two exponential integrals which I don't know how to evaluate. $\endgroup$ – Tyg13 May 12 '15 at 18:34
  • $\begingroup$ For the second one, I did not think you should come to that integral, but since you did, you could look up Frullani's integral. For the first one, I won't give it away already... Try again :) $\endgroup$ – mickep May 12 '15 at 18:50
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Let $P$ be some polynomial. What is the primitive of $e^xP(x)$?

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    $\begingroup$ For a polynomial with degree $k$, I used integration by parts to find $I = e^{x}[P(x) - P'(x) + \ldots + (-1)^k P^{(k)}(x)] = e^{x}\sum\limits_{n=0}^{k} (-1)^{n} P^{(n)}(x)$. Then, I tried to deduce a relation for $P^{(n)}(x)$ by representing it as a power series $P(x) = \sum\limits_{n=0}^{k} a_{n}x^{n}$ and finding the general formula for differentiation as $\sum\limits_{i=n}^{k} a_{i}x^{i-n} = a_{n} + a_{n+1} + \ldots + a_{k}x^{k-n}$ The overall integral $I = e^{x} \sum\limits_{n=0}^{k}\left(\sum\limits_{i=n}^{k}a_{i}x^{i-n}\right)(-1)^{n}$, but I don't know if this can be further simplified $\endgroup$ – Tyg13 May 12 '15 at 3:01
  • $\begingroup$ Yes, this is correct. To formally proof it you should use induction. To my knowledge, it cannot be simplified further. $\endgroup$ – wythagoras May 12 '15 at 11:42
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I am not exactly sure if this is at the multivariable calculus level you are looking for - perhaps too easy, perhaps too difficult. Anyhow, perhaps it is interesting, especially if you like symmetry:

Determine the integral $$ \int_0 ^1 \int_0 ^1 { \frac{(xy)^k}{1 + \left( \frac{y}{x} \right)^p} } \mathrm{d} x \mathrm{d} y. $$

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This is a great example of using integration along with a little summation skills. Try and find a closed form of \begin{equation} \Gamma(n,k)=\int_0^1 (-\log x)^{k-1}x^{n-1}dx \end{equation} using integration by parts.

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Physically-motivated problems include centers of mass and moments of inertia for solid balls, spherical shells, ellipsoids, rectangular blocks, cylinders, cones, and circular tori (with various density functions and about various axes). (Answers, assuming constant density.)

Depending on calibration, all of these may be "too easy", but there's the compensation of computing quantities with "real-world significance"....

Calculating the gravitational potential for the region between two concentric solid balls (assuming constant density) is pleasantly challenging. (And then once you have the answer for a solid ball, you can work out an old favorite: A planet is a solid ball of constant density. A straight tunnel is bored between two points $A$ and $B$ on the surface, and an object released at $A$ moves through the tunnel under the force of gravity (without friction). Show that the time of arrival at $B$ does not depend on $A$ and $B$, i.e., depends only on the mass and radius of the planet. That is, if the earth were a ball of constant density, and if straight tunnels were drilled from, say, New York to Toronto, Paris, Lagos, Rio de Janeiro, Mumbai, and Perth, then an object dropped into any of these tunnels would reach the other end in the same amount of time, assuming no friction. If memory serves, the travel time for the earth would be about 40 minutes.)

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Find the following sum:

$$f(a) = \sum_{n=0}^{\infty} \int_a^{\infty} x e^{-nx} dx$$

for $a>0$. Note: You will need a polylogarithm.

If this was to easy, take $x^2$ or $x^3$ instead.

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