Question

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

Answer 1

`ln(lnx)`

Explanation:

antiderivative of `1/(xlnx)` could be written as `int(1/(xlnx)dx)`

Or as `int1/(lnx)*1/xdx`

Recall that : the derivative of `lnx` is `1/x`
This means that : `(d(lnx))/(dx)=1/x`

Implying : `d(lnx)=1/x(dx)`

Hence, `int1/lnx*1/xdx=int1/lnx*d(lnx)`

we are left with the antiderivative of `1/lnx` with respect to `lnx`

This case is simlar to finding the antiderivative of say, `1/u` wrt `u`
The answer would be `lnu`

Similarly, the antiderivative of `1/lnx` with respect to `lnx` is `color(blue)(ln(lnx))`