Q$4(5$ points$)$

Let $a,$binR. Suppose that $f$ is invertible and $f^{\text{'}}$ is continuous on $a,b.$ Show the following statements are true.

$($i$){\displaystyle \underset{a}{\overset{b}{}}}x{f}^{\text{'}}(x)dx={\displaystyle \underset{f(a)}{\overset{f(b)}{}}}{f}^{-1}(w)dw.$

$($ii$){\displaystyle \underset{a}{\overset{b}{}}}f(x)dx=bf(b)-af(a)-{\displaystyle \underset{f(a)}{\overset{f(b)}{}}}{f}^{-1}(x)dx.$

Hint: Use the substitution rule in $($i$)$ and integration by parts in $($ii$).$