Antiderivatives

Question

I have this definite integral and I need to find its derivative with respect to $t$. I am not sure how to do it without computing the value of the integral and taking its derivative. $$\int_2^t \sqrt{u^4-16}\;du$$

Answer 1

Apply Fundamental theorem of Calculus here. Let's define $$f(t)=\int_0^t\sqrt{u^4-16} du.$$ Then $f'(t)=\sqrt{t^4-16}$.

Answer 2

HINT

Note that in general

$$f(t)=\int_{a(t)}^{b(t)}g(u) du\implies f'(t)=g(b(t))\cdot b'(t)-g(a(t))\cdot a'(t)$$