Puzzling Integral Solutions

At the falls clubs fair for the Mathematical Sciences Society at University of Alberta, we put up a poster board with two challenging integrals, offering free t-shirts to anybody who solved them. We put out several mini-whiteboards for people to work on, and to my surprise, there was quite a bit of interest (free tshirts are always a great motivator!). Here’s the solutions to two of these challenges. I really like how the first one looks symbolically challenging, but conceptually is not, while the second one looks more elementary, but involves a clever trick courtesy of Richard Feynman.

Link to Integral Solutions