How do you use the fundamental theorem of calculus to find F'(x) given $F (x) = \int t^{2} \sqrt{1 - t^{3}} d t$ $F(x)=int t^2sqrt(1-t^3)dt$ from [0,x]?

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Answer 1 · 2017-03-03T22:08:31

$F'(x) = x^2sqrt(1-x^3)$

If asked to find the derivative of an integral using the fundamental theorem of Calculus, you should not evaluate the integral

$d/dx \ int_a^x \ f(t) \ dt = f(x)$

(ie the derivative of an integral gives us the original function back).

We are asked to find:

$F'(x) = d/dx F(x)$
$" " = d/dx \ int_0^x \ t^2sqrt(1-t^3) \ dt$
$" " = f(x)$ , where $f(t)=t^2sqrt(1-t^3) \ \ \$ (using FTOC)
$" " = x^2sqrt(1-x^3)$

How do you use the fundamental theorem of calculus to find F'(x) given F(x)=int t^2sqrt(1-t^3)dtF(x)=∫t2√1−t3dtF(x)=int t^2sqrt(1-t^3)dt from [0,x]?