The integral ∫ ( ((x)/(2)) ^ x + ((2)/(x)) ^ x ) _ 2 xdx is equal to

Given,

$I = ∫ \left({\left(\dfrac{x}{2}\right)}^{x} + {\left(\dfrac{2}{x}\right)}^{x}\right)\log_{2}xdx$

Now, Let ${\left(\dfrac{x}{2}\right)}^{x} = t$

$⇒ x{\log}_{2}\left(\dfrac{x}{2}\right) = \log_{2}t$

$⇒ x{\log}_{e}\left(\dfrac{x}{2}\right) \cdot \log_{2}e = \log_{e}t \cdot \log_{2}e$

On differentiating both sides we get :
$\log_{e}\left(\dfrac{x}{2}\right) + x \cdot \dfrac{2}{x} \cdot \dfrac{1}{2} = \dfrac{1}{t}\dfrac{dt}{dx}$

$⇒ \log_{e}\left(\dfrac{x}{2}\right) + 1 = \dfrac{1}{t}\dfrac{dt}{dx}$

Then solution is not possible as there is no proper substitution.

Note: This question was bonus in Jee Mains 2023 April session.