Where to find interesting integrals for a Calc III student? 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.
 A: 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 $a=1$.

2) Let $a>-1$ and $b>-1$. Calculate the integral
$$
\int_0^1\frac{x^b-x^a}{\ln x}\,dx.
$$
A: Let $P$ be some polynomial. What is the primitive of $e^xP(x)$?
A: 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
$$
\int_0 ^1 \int_0 ^1 { \frac{(xy)^k}{1 + \left( \frac{y}{x} \right)^p} } \mathrm{d} x \mathrm{d} y.
$$
A: 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.
A: 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).
A: 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.)
A: 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. 
