How do you integrate ((ln(x))^7)/x dx?

1 Answer
Mar 14, 2018

frac(1)(8) (ln(x))^(8) + C

Explanation:

We have: int frac((ln(x))^(7))(x) dx

= int (ln(x))^(7) cdot frac(1)(x) dx

Let u = ln(x) Rightarrow du = frac(1)(x) dx:

= int u^(7) du

= frac(u^(7 + 1))(7 + 1) + C

= frac(1)(8) u^(8) + C

Replace u with ln(x):

= frac(1)(8) (ln(x))^(8) + C