If the integral $I =\int\left\{\log_e(\log_ex)^2+\frac{α}{\log_ex}\right\}dx=x\log_e(log_ex)^2+C$, where C is constant of integration. Then the value of $α$ is:

2

The correct answer is Option (4) → 2

Given

$\int\left\{\log_e(\log_e x)^2+\frac{\alpha}{\log_e x}\right\}dx = x\log_e(\log_e x)^2+C$

Differentiate RHS

$\frac{d}{dx}\left[x\log_e(\log_e x)^2\right]$

$=\log_e(\log_e x)^2 + x\cdot \frac{d}{dx}\left[\log_e(\log_e x)^2\right]$

$\frac{d}{dx}\left[\log_e(\log_e x)^2\right] =\frac{1}{(\log_e x)^2}\cdot 2\log_e x \cdot \frac{1}{x} =\frac{2}{x\log_e x}$

Hence

$\frac{d}{dx}\left[x\log_e(\log_e x)^2\right] =\log_e(\log_e x)^2+\frac{2}{\log_e x}$

Compare with integrand

$\log_e(\log_e x)^2+\frac{\alpha}{\log_e x}$

$\alpha=2$

The value of $\alpha$ is $2$.