Question

How do you use part 1 of the Fundamental Theorem of Calculus to find the derivative of `y=int(sqrtt sintdt)` from `[x^3, sqrtx]`?

Answer 1

You need one constant limit of integration to apply FTC Part 1.

So rewrite the integral:

`y=int_(x^3)^sqrtx (sqrtt sintdt) = int_(x^3)^1 (sqrtt sintdt) +int_1^sqrtx (sqrtt sintdt)` ,

So `y = - int_1^(x^3) (sqrtt sintdt) +int_1^sqrtx (sqrtt sintdt)`

Now use FTC 1 and the Chain Rule to get:

`-sqrt(x^3)sin(x^3)3x^2 + sqrtsqrtx sin (sqrtx) 1/2sqrtx`

`= -3x^3sqrtxsin(x^3) + root(4)x / (2sqrtx) sin(sqrtx)`

Or whatever rewrite you favor as "simplified".