Q.Choose the correct option: $\int \frac{\log x}{(\log ex)^2}dx =$
(A) $\frac{x}{1+\log x}+c$ (B) $x(1+\log x)+c$ (C) $\frac{1}{1+\log x}+c$ (D) $\frac{1}{1-\log x}+c$

Since $\log(ex)=\log e+\log x=1+\log x$, the integrand is $\frac{\log x}{(1+\log x)^2}$. Check the derivative of the candidate $\frac{x}{1+\log x}$ using the quotient rule: $\frac{d}{dx}\left[\frac{x}{1+\log x}\right]=\frac{(1+\log x)\cdot1-x\cdot\frac1x}{(1+\log x)^2}=\frac{1+\log x-1}{(1+\log x)^2}=\frac{\log x}{(1+\log x)^2}$, …

Simplify $\log(ex)$, then recognize the integrand as an exac …

Forgetting to expand $\log(ex)$ into $1+\log x$ and instead treating $\log(ex)$ and $\log x$ as unrelat …