Question about integrating to find length of a polynomial line between two points

Question

I came across this integral when finding the length of a line: $$\int_{0}^2 \sqrt{1+(16t^3-36t^2+24)^2} dt$$ and I realized that if $16t^2-36t^2+24$ was a single variable such as x, you could use trig substitution to compute the integral. So I tried manipulating the integral by substituting $u = 16t^3 - 36t^2 + 24$, and other crazy things such as finding two perfect squares that multiply to get $1+(16t^3 -36t^2 + 24)$, in order to get rid of the square root. I worked on this for hours and could not find a solution to my problem. Any help would be appreciated, as I am a student that loves math, and I want to learn.

Answer 1

Not sure if this is what was asked, but you can't solve this integral "the traditional way"

However you can do it numerically. Try the Clenshaw-Curtis method:

https://link.springer.com/article/10.1007/BF01403676

I tried using a python lib called Scipy to solve this, and I got the answer $18.657011670468133$ Scipy uses the Clenshaw-Curtis method by default, so you should be on the right track with it. I hope this will get you closer to the solution :D

P.S Wolfram gets $18.657011670468133$ too