Real Analysis: Integration by parts

[sweetie5985]

I'm trying to prove integration by parts rigorously.

Suppose f: [a,b] -> Reals, g: [a,b] -> Reals are differentiable functions and f ' and g' are continuous on [a,b].

Want to show: Integral from [a,b] of f(t)g'(t)dt = f(b)g(b) - f(a)g(a) - integral from [a,b] of f '(t)g(t)dt

Let h(x) = integral from [a,x] of f(t)g'(t0
Let k(x) = f(x)g(x)-f(a)g(a) - integral from [a,x] of f '(t)g(t)dt

I know I need to use these two functions to prove integration by parts rigorously but I don't know where to go from here.

 

[daon]

Well...

\(\displaystyle \int_a^b \left( \frac{d}{dx} \int_k^x \left(f'(t)g(t)+g'(t)f(t) \right )dt \right ) dx = \int_a^b (fg)' dx\)