Question

Let F(x) = integral of 2t/t^2+1 from 2x to x^2, find the value of Fprime(2)? (Picture included)

Answer 1

`F'(2) = 16/17`

Explanation:

`F(x) = int_(2x)^(x^2) (2t)/(t^2 + 1) \ dt = int_(2x)^(x^2) f(t) \ dt`

Liebnitz Integral Rule:

`F'(x) = f(x^2) (x^2)^' - f(2x) (2x)^'`

`= (2(x^2))/((x^2)^2 + 1) * 2x - (2(2x))/((2x)^2 + 1) * 2`

`= (4 x^3)/(x^4 + 1) - (8 x)/(4 x^2 + 1)`

`F'(2) = 16/17`