How to solve an integral of the function: $f(x) = \ln (x) / (x \sqrt{x})$

Question

By the way the the question is in title. It seems this might be solved by substituting, but I can't make out what to do with the root.

Answer 1

$$\int \frac{\ln(x)}{x\sqrt{x}}\text{d}x$$

Set $\ln(x) = t\quad \text{thence}\quad \text{d}t = \frac{\text{d}x}{x}$ and also $x = e^{t}$, so that

$$\int \frac{t}{e^{t/2}}\ \text{d}t = \int t e^{-t/2}\ \text{d}t = -2e^{-t/2}(2+t)$$

Substituting back to x you get $$-\frac{2 (\log (x)+2)}{\sqrt{x}}$$

Answer 2

Hint: note that $d(\frac{1}{\sqrt x})=-\frac{1}{2x\sqrt x}$ and $\ln x=-2\ln \frac{1}{\sqrt x}$. Can you put it together?