TITLE: Integral that evaluates to 42
QUESTION [11 upvotes]: I know this isn't a specific math question, but I need some help coming up with an integral that evaluates to 42 for a calc shirt. Obviously, I'm striving for something more complicated than $\int_{10}^{11} 4x dx$ or $\int_{0}^{42} 1 dx$. Try and stick to integral content that would be covered in the BC calc curriculum, because that is the class I'm trying to design the shirt for. Thanks!

REPLY [15 votes]: Because $42$ is the fifth Catalan number then using the known integral representation of the Catalan's numbers we have that
$$C_5=\frac1{2\pi}\int_0^4x^5\sqrt{\frac{4-x}x}\,\mathrm dx=42$$
Playing with some change of variable above you can get different integral expressions. By example with the change of variable $(4-x)/x=t$ we can build the following improper integral
$$C_5=\frac{16}{\pi}\int_0^\infty\sqrt t\left(\frac12+\frac{t}2\right)^{-7}\mathrm dt=42$$

Maybe more interesting integral representations can be achieved considering first the integral representations for $21$, what is a number with more interesting properties.