Hi, I follow this channel and I love it!
During my PhD I came across this nice integral which I believe is a cool challenge:
Subject to the conditions
The integral is actually somewhat easy once you know the trick. The result is really nice since it is used to compute an upper bound for the settling time of an interesting differential equation independently to the initial condition. This is, the solution of such differential equation converges to the origin before the value computed by the integral no matter how far from the origin it starts. This has many applications in robotics, computer science and control theory since it allows you to design algorithms that "finish" their work before a deadline, no matter how bad their initial guess for the solution was.
All details, summary of applications and the solution of the integral can be found here:
However, it would be more fun not to spoil the solution before trying to do the integral first!
(The solution is in Proposition 1)
HINT: one may start with k=1, p=1-s and q=1+s with 0<s<1 which is a restricted case, but motivates the general "trick" to solve integral.
Thanks for all your videos!