Question

Evaluate $\int \frac{\log \left(x/e\right)}{(\log x{)}^2}dx$

• A. $\frac{\log x}{x}+c$
• B. $\frac{x}{\log x}+c$
• C. ${}^{\frac{x}{log(x{)}^2}+c}$
• D. $\frac{(\log x{)}^2}{x}+c$

Answer 1

The correct option is B $\frac{x}{\log x}+c$

$\int \frac{\log \left(x/e\right)}{(\log x{)}^2}dx=\int \frac{\log x-1}{(\log x{)}^2}$

$=\int [\frac{1}{\left(\log x\right)}-\frac{1}{(\log x{)}^2}]dx$

$=\int \frac{1}{\left(\log x\right)}dx-\int \frac{1}{(\log x{)}^2}dx$

$=\frac{1}{\left(\log x\right)}\int 1.dx+\int \frac{1}{(\log x{)}^2}dx-\int \frac{1}{(\log x{)}^2}dx$ ..... [Using integration by parts]

$=\frac{x}{\log x}+c$

$\therefore \int \frac{\log \left(x/e\right)}{(\log x{)}^2}dx=\frac{x}{\log x}+c$