Question

Not sure how to approach this one, any help would be appreciated!

If
`f'` is continuous on
`[0, \infty)` and
`\lim_(x \to \infty) f(x) =0` , show that
`\int\limits^\infty_0 {f'(x)} \, dx =-f(0)` .

Answer 1

The continuity of
`f'` and its limiting behavior guarantees that
`f'` is Riemann integrable, so you can write


`\displaystyle\int_0^\infty f'(x)\,\mathrm dx=uv\bigg|_(x=0)^(x\to\infty)-\int_0^\infty v\,\mathrm du`

where
`u=1\implies\mathrm du=0\,\mathrm dx` and
`\mathrm dv=f'(x)\,\mathrm dx\implies v=f(x)`, so that


`\displaystyle\int_0^\infty f'(x)\,\mathrm dx=f(x)\bigg|_(x=0)^(x\to\infty)=\lim_(x\to\infty)f(x)-f(0)=-f(0)`