Integral of a continuously differentiable function on [a,b].

[rsa58]

suppose f is real, continuously differentiable function on [a,b], f(a)=f(b)=0 and

integral f^2dx=1

show [integral(xf(x)f'(x)dx)= -1/2 over [a,b]

 

[Rainbow Child]

How can you write the term $$f(x)\,f'(x)$$?

 

[mathman]

suppose f is real, continuously differentiable function on [a,b], f(a)=f(b)=0 and

integral f^2dx=1

show [integral(xf(x)f'(x)dx)= -1/2 over [a,b]

Integrate by parts:

You will get 1/2(bf²(b)-af²(a)-integral(a,b)f²(x))

 

[HallsofIvy]

Elaborating on Rainbow Child's suggestion, what is the derivative of (f(x))²?