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

##### 1 Answer

Sep 26, 2017

# 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 #