How to find the indefinite integral for a natural log being divided by x?

Question

I've done many examples in the math book but none of them have a natural log as a numerator. Here's the question.

$$\int\frac{(\ln\ x)^7}{x} dx$$

I am given these 2 properties, where $u$ is a function.

$$\int\frac{1}{x}dx = \ln|\ x\ |+C$$

$$\int\frac{u'}{u}du = \ln|\ u\ |+C$$

Answer 1

Do the substitution $u=\ln x$. This gives us $\mathrm{d}u= \dfrac1x \ \mathrm{d}x$. Replacing $\dfrac1x \ \mathrm{d}x$ with $\mathrm{d}u$ gives us $$\int\! u^7 \ \mathrm{d}u=\frac{u^8}{8}+C$$ Back substituting $\ln x=u$ $$\frac{(\ln x)^8}{8}+C$$

Answer 2

by using the by parts method $$\int\frac{(\ln\ x)^7}{x} dx=(\ln x)^7.\ln x-7\int\frac{(\ln\ x)^7}{x} dx+C$$ $$8\int\frac{(\ln\ x)^7}{x} dx=(\ln x)^8+C$$ $$\int\frac{(\ln\ x)^7}{x} dx=\frac{(\ln x)^8}{8}+K$$