Question

Use substitution to evaluate the indefinite integral:

The integral of [ (sq. root (1 + ln x)) ((ln x)/x) dx]

Im confused on what i should substitute u and du for.

Thank you so much!!

Answer 1

Let u = 1+ln x
so,
du = 1/x dx
ln x = u-1

So, we have

u^1/2 * (u-1) du
= u^3/2 - u^1/2 du

Integrate that to get

2/5 u^5/2 - 2/3 u^3/2
=
2/5 (1 + ln x)^5/2 - 2/3 (1+ln x)^3/2

Check your work by taking the derivative. Trust me -- it comes out right.