Question

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

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

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