Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous, diffrentiable given function. Find $\int^b_af^{(3)}(x)\ f'(x)dx$ :
Hey everyone. I've tried integration by parts: let $u'= f^{(3)}(x)\Rightarrow u=f''(x), y=f'(x)\Rightarrow y'=f''(x)$ We get- $\int^b_af^{(3)}(x)\ f'(x)dx = $
$ f''(b)\ f'(b)-f''(a)\ f'(a)- \int^b_af''(x)\ f''(x)dx$ $\bigg [u=f''(x)\Rightarrow u'=f^{(3)}(x) , y'=f''(x)\Rightarrow y=f'(x)\bigg ]$
$= f''(b)\ f'(b)-f''(a)\ f'(a)- f''(b)\ f'(b)+f''(a)\ f'(a)- \int^b_a f^{(3)}(x)\ f'(x)dx $
Hence, I get: $ 2\int^b_af^{(3)}(x)\ f'(x)dx=0 $
Does that make sense? Thanks in advance.