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

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Answer 1 · 2018-03-14T21:24:05

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

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#