Step 1: Choose integration by parts:
Neither $\int\dfrac{dx}{\ln x}$ nor $\int\dfrac{dx}{(\ln x)^2}$ has an elementary form alone, but they are linked. Integrate $\int\dfrac{dx}{\ln x}$ by parts with $u=\dfrac1{\ln x}$ and $dv=dx$.
Step 2: Do the parts:
$\int\dfrac{dx}{\ln x}=\dfrac{x}{\ln x}+\int\dfrac{dx}{(\ln x)^2}$. Therefore $\int\left[\dfrac1{\ln x}-\dfrac1{(\ln x)^2}\right]dx=\dfrac{x}{\ln x}$.
Step 3: Evaluate:
From 2 to $e$: $e-\dfrac{2}{\ln2}$, so $a=e$, $b=-2$. Option C.
Final Answer:
The integrand is the derivative of x / ln x, giving e - 2/ln 2.
\[ \boxed{\text{(C) }a=e,\ b=-2} \]