Question

Find `int \ ln(sqrt(x))/x dx`?

Answer 1

`int \ ln(sqrt(x))/x \ dx = ln^2x/x + C`

Explanation:

We seek:

`I = int \ ln(sqrt(x))/x \ dx`

Using the properties of logarithms, we can write this integral as follows:

`I = int \ 1/2 \ ln(x)/x \ dx`

Which then leads to a suggestive substitution:

Let `u =ln x` => `(du)/dx = 1/x`

So if we now substitute this into the integral, we get:

`I = 1/2 \ int \ u \ du`

Which is now a standard integral, so we have:

`I = 1/2 u^2/2 + C`

And reversing the substitution:

`I = 1/4 (lnx)^2 + C`