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:
Answer & explanation
Correct answer: option 4
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 $2$.