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 $\int\limits_{1}^{5} f(x) d x+\int\limits_{2}^{10} g(y) d y$ equals

Answer 1

$y=f(x) \Rightarrow x=f^{-1}(y)=g(y)$
$d y=f'(x) d x$
$\therefore I=\int\limits_{1}^{5} f(x) d x+\int\limits_{1}^{5} x f'(x) d x$

where $y$ is $2$ then $x=1$
$y$ is $10$ then $x=5$
$\therefore I=\int\limits_{1}^{5}\left(f(x)+x f'(x)\right) d x$
$=\left.x f(x)\right|_{1} ^{5}=5 f(5)-f(1)=5 \cdot 10-2=48$