Question

What is `int (lnsqrt(3x+2))^2dx`?

Answer 1

Here is some help to get you started.

Explanation:

`int (lnsqrt(3x+2))^2dx`

Let's look at the integrand for a bit:

`(lnsqrt(3x+2))^2`

First note that `(lnsqrt(3x+2)) = ln(3x+2)^(1/2) = 1/2ln(3x+2)`

and `3x+2` is linear, so we can do a substitution.

Then the question will be how to find

`int (lnt)^2 dt`

Details

`int(lnsqrt(3x+2))^2dx = int (1/2ln(3x+2))^2 dx`

`= 1/4 int (ln(3x+2))^2 dx`

With `t = 3x+2`, the integral becomes:

`= 1/12 int (lnt)^2 dt`

Now try substitution, try parts, try anything else you think of.

Hint
Recall that we can integrate `lnx` by using integration by parts.