How do you calculate the derivative of #int(2t-1)^3 dt # from #t=x^2# to #t=x^7#?
We also know that
so evaluating the derivative of that integral is exactly the same as evaluating
Note that further simplification is applicable (but not required) since this is a difference of cubes.
We can use the Fundamental Theorem of Calculus , Part 1:
But if it is:
To make this particular integral fit within the FTC, we need also do some interval manipulation:
Splitting out the integration intervals:
Switching the first interval limits:
Applying the FTC:
Maybe that can be simplified a bit more.