Question

How do you integrate `1/(xlnx)dx`?

Answer 1

Hello !

I propose another solution.

Remember that `(\ln(u))' = \frac{u'}{u}` if `u` is a positive differentiable function.

Take `u (x) = \ln(x)` for `x>1` : it's a positive differentiable function.

Remark that `\frac{u'(x)}{u(x)} = \frac{\frac{1}{x}}{\ln(x)} = \frac{1}{x\ln(x)}`, then

`\int \frac{\text{d}x}{x\ln(x)} = \ln(u(x)) + c = \ln(\ln(x)) + c`,

where `c` is a real constant.