Question

Find the integral

Answer 1

have you tried integration by parts?

Answer 2

make x^(1/2) your integrating term
make ln(x) your differentiating term

Answer 3

\(\color{#000000 }{ \displaystyle \int \sqrt{x}\ln(x)~dx }\)

\(\color{#000000 }{ \displaystyle \frac{2}{3}\int \sqrt{x}\ln(x^{3/2})~dx }\)

\(\color{#000000 }{ \displaystyle u=x^{3/2} }\)
\(\color{#000000 }{ \displaystyle du= \frac{3}{2}x^{1/2}~dx\quad \Longrightarrow \quad \frac{2}{3}du= x^{1/2}~dx }\)

\(\color{#000000 }{ \displaystyle \frac{4}{9}\int \ln(u)~du }\)

That would be, if ln(u) is an automatic integral for you...
In general, myininaya's approach is better...

Answer 4

Just looking at a different angle.

Answer 5

that is really good looking @SolomonZelman
one can skip the integration by parts if they are allowed to remember the integral of ln(u) w.r.t. u

Answer 6

Yes, that would be a nice approach\(: ) \) Of course one can derive the general antiderivative of \(x^n\ln(x)\), using this trick.