Urgent, The Fundamental Theorem of Calculus !

Question

Use the Fundamental Theorem of Calculus to find the derivative of 
f(x)= ∫ ((t^2/4)-1)^5 dt, upperbound-x^2, lowerbound= 4

Answer 1

Since the lower limit is a constant, the Fundamental Theorem of Calculus says that, to find the derivative of the integral, replace t in the integrand by the upper limit and then multiply that function by the derivative of the upper limit.

 

So, the answer is:   [(x²)²/4 - 1]⁵(2x) = 2x(x⁴/4 - 1)⁵.

Answer 2

One version of the Fundamental Theorem is that the derivative of an expression like

 

    F(x) =  ∫ f(t) dt        with upper bound   u(x) and lower bound v(x)  is

 

     F' (x) =   f(x) [ u' - v' ]   where the prime denotes derivative.

 

   This is a lot like the chain rule.

 

  For your problem,    u(x) =  x²   and v(x) = 4.       So   u'  = 2x  and v' = 0

 

 and  the answer is   (x²/4)⁵ 2x