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

Answer 1

The integrand is of the form u du where u=f(x) and du=f'(x) dx.

The integral is u²/2, i.e. f²/2.

Now evaluate the integral at the upper and lower limits...and don't forget the factor of 2 at the outside of the integral sign!