Question

What is the antiderivative of ln(x)/x?

Answer 1

`(ln(x))^2/2+C` (Also written `1/2ln^2(x)+c`).

Explanation:

`ln(x)/x=ln(x)*(1/x)`.

The derivative of `lnx` is `1/x`, so

`ln(x)*(1/x)` is of the form: `f(x)*f'(x)`, so the antiderivative is `1/2 (f(x))^2 +C`.

(Alt notation: `ln(x)/x=ln(x)*(1/x)` is of the form `u (du)/(dx)` whose antiderivative is `u^2/2 +C`.)