Question

If $J=\int \frac{x+y}{xy}dy\text{and}I=\int \frac{x+y}{x^2}dx$, where $x$ and $y$ are independent variables such that $g\left(\frac{y}{x}\right)=J-I\text{and}g\left(1\right)=1,$ then value of $\left[g\right(e\left)\right]$ is
where [.] denotes the greatest integer function

Answer 1

$J=\int (\frac{1}{y}+\frac{1}{x})dy=lny+\frac{y}{x}+C_{1}I=\int (\frac{1}{x}+\frac{1}{x^2})dx=lnx-\frac{y}{x}+C_{2}g\left(\frac{y}{x}\right)=ln\left(\frac{y}{x}\right)+\frac{2y}{x}+c\frac{y}{x}=1\Rightarrow g\left(1\right)=1=ln\left(1\right)+2+c\Rightarrow c=-1\frac{y}{x}=e\Rightarrow g\left(e\right)=lne+2e-1=2e=5.42/5.43/5.44$
$\Rightarrow \left[g\right(e\left)\right]=5$