Question

How do I prove that this integral from a to b is equivalent to the second statement?

If f' is continuous on [a, b], show that `2int_a^bf(x)f'(x)dx)= [f(b)]^2-[f(a)]^2`

Answer 1

`2 int_a^b \ f(x) \ f'(x) \ \ dx`

`= 2 int_a^b \ d/(dx)(1/2 f(x) )^2 \ \ dx`

`= int_a^b \ d/(dx)( f(x) )^2 \ \ dx`

`= [(f(x))^2]_a^b`

`= (f(b))^2-(f(a))^2`