Question

Let $f(x)=12\left(\frac{e^{3 x}-3 e^{x}}{e^{2 x}-1}\right)$ be defined for $x>0$ and $g(x)$ be the inverse of $f(x)$ If $\int\limits_{8}^{27} g(x) d x=a \ln 3-b \ln 2-c$, then the value of $a-(b$ $+c)$ is

Answer 1

$\int\limits_{\ln 2}^{\ln 3} f(x) d x+\int\limits_{8}^{27} g(y) d y=27 \ln 3-8 \ln 2$
and $\int\limits_{\ln 2}^{\ln 3} f(x) d x=12-12 \ln 3+12 \ln 2$
$\therefore \int\limits_{8}^{27} g(y) d y=39 \ln 3-20 \ln 2-12$
Hence, $a=39 ; b=20 ; c=12$
$\therefore a-(b +c)=39-32=7 .$