Question

Which of the following represents the integral ${\int \frac{1}{x\ln{x}}dx}$

1) $ln(lnx)+C$

2) $ln(xlnx)+C$

3) $lnxlnx+C$

4) None of the above

Answer 1

$\smallint\frac{1}{xln(x)}dx$

Substitute $u=ln(x)$

Hence $\frac{du}{dx}=\frac{1}{x},$ $dx=xdu.$

Therefore, = $\smallint\frac{1}{u}du$

This the standard integral;

$=ln(u)+C$

Undo substitution $u=ln(x)$

$=ln(ln(x))+C$

Hence the answer is $ln(lnx)+C$