Indefinite Integrals Using Natural log

Question

Hi can someone please help?

I need to evaluate this indefinite integral:

$$\int \frac{(\ln x)^5}x dx$$

I know I need to use substitution, so if I let u= x but I can't figure out the antiderivative for the top portion.

Thank you!

Answer 1

Let $\ln x=t\implies \frac{1}{x}\ dx=dt$ $$\int \frac{(\ln x)^5}{x}\ dx=\int t^5\ dt$$$$=\frac{t^6}{6}+C$$ substituting $t=\ln x$, $$=\frac{(\ln x)^6}{6}+C$$

Answer 2

Let $u=\ln x$ and you get...

$$\int u^5 du$$

$$=u^6/6+C$$ $$=\frac{(\ln x)^6}{6}+C$$