JEE 2023 — Mathematics PYQ

Let $I=\int \left[{\left(\frac{x}{2}\right)}^x+{\left(\frac{2}{x}\right)}^x\right]\ln \left(\frac{ex}{2}\right)dx$

$=\int \left[e^{x\ln x-x\ln 2}+e^{x\ln 2-x\ln x}\right]dx$

Let $x\ln x-x\ln 2=t$

$(\ln x+1-\ln 2)dx=dt$

$\Rightarrow \ln \left(\frac{ex}{2}\right)dx=dt$

$\therefore I=\int \left(e^t+e^{-t}\right)dt$

$=e^t-e^{-t}+c$

$={\left(\frac{x}{2}\right)}^x+{\left(\frac{2}{x}\right)}^x+c$