Question

If $g(x)$ is the inverse of $f(x)$ and $f(x)$ has domain $x\in$ $[1,5]$, where $f(1)=2$ and $f(5)=10$ then the values of $\underset{1}{\overset{5}{\int }}f(x)dx+\underset{2}{\overset{10}{\int }}g(y)dy$ equals

Answer 1

Answer: 48

$y=f(x)\Rightarrow x=f^{-1}(y)=g(y)$
$dy=f^{\prime }(x)dx$
$\therefore I=\underset{1}{\overset{5}{\int }}f(x)dx+\underset{1}{\overset{5}{\int }}x{f}^{\prime }(x)dx$

where $y$ is $2$ then $x=1$
$y$ is $10$ then $x=5$
$\therefore I=\underset{1}{\overset{5}{\int }}\left(f(x)+x{f}^{\prime }(x)\right)dx$
$={xf(x)|}_1^5=5f(5)-f(1)=5\cdot 10-2=48$