Cool Integral

I am a fan of the show and would love to see you cover this in a video. I teach calculus and came up with a fun problem for my students:

The answer to this integral is 5pi^2/12. The trick is to write it like this:

and then notice that this is the end result of a double integral

You can swap the order of integration and complete the square

From here, the method to use depends on the values of a and b. If |y| < 2, then the integral is an arctan. If y > 2, then we'd have to do partial fraction decomposition and integrate to logs. I'll show the first one and just state the answer for the second. Assuming the first case, we get:

Then trig sub y = 2sin(theta), the integral becomes

since arcsin(x)+arccos(x) = pi/2. The final solution is

for |a|, |b| < 2. For 2 < a < b, the answer is

by doing hyperbolic substitution on the other case. I hope this was a fun trick to see!