question_answer

Evaluate \[\int{\frac{1}{x{{(\log \,x)}^{n}}}\,dx.}\]

Answer:

Let \[l=\int{\frac{1}{x\,{{(\log x)}^{n}}}dx}\] Put                   \[t=\log x\] \[\Rightarrow \]               \[dt=\frac{1}{x}dx\] \[\therefore \]      \[l=\int{\frac{1}{{{(t)}^{n}}}dt=\frac{{{t}^{-\,n+1}}}{-\,n+1}}+C\] \[=\frac{{{(\log x)}^{1\,\,-\,\,n}}}{1-n}+C\]                       \[[\because t=\log x]\]