# What is the antiderivative of  1/(xlnx)?

Sep 26, 2015

$\ln \left(\ln x\right)$

#### Explanation:

antiderivative of $\frac{1}{x \ln x}$ could be written as $\int \left(\frac{1}{x \ln x} \mathrm{dx}\right)$

Or as $\int \frac{1}{\ln x} \cdot \frac{1}{x} \mathrm{dx}$

Recall that : the derivative of $\ln x$ is $\frac{1}{x}$
This means that : $\frac{d \left(\ln x\right)}{\mathrm{dx}} = \frac{1}{x}$

Implying : $d \left(\ln x\right) = \frac{1}{x} \left(\mathrm{dx}\right)$

Hence, $\int \frac{1}{\ln} x \cdot \frac{1}{x} \mathrm{dx} = \int \frac{1}{\ln} x \cdot d \left(\ln x\right)$

we are left with the antiderivative of $\frac{1}{\ln} x$ with respect to $\ln x$

This case is simlar to finding the antiderivative of say, $\frac{1}{u}$ wrt $u$
The answer would be $\ln u$

Similarly, the antiderivative of $\frac{1}{\ln} x$ with respect to $\ln x$ is $\textcolor{b l u e}{\ln \left(\ln x\right)}$