Question

$\int \frac{\left[\log \right(\log x){]}^m}{x\log x}dx=$

• A. $\frac{{\left[\log \left(\log \left(x\right)\right)\right]}^{m+1}}{m+1}+c$
• B. $\frac{{\left[\log \left(\log \left(x\right)\right)\right]}^m}{m}+c$
• C. $\frac{{\left[\log \left(\log \left(x\right)\right)\right]}^{m+1}}{m}+c$
• D. $\frac{{\left[\log \left(\log \left(x\right)\right)\right]}^m}{m+1}+c$

Answer 1

The correct option is A $\frac{{\left[\log \left(\log \left(x\right)\right)\right]}^{m+1}}{m+1}+c$

$I=\int \frac{\left[\log \right(\log x){]}^m}{x\log x}dx$

Put $\log \left(\log x\right)=t$

$\frac{1}{x\log x}dx=dt$

$I=\int t^{m}dt$

$=\frac{t^{m+1}}{m+1}$

$I=\frac{{\left[\log \left(\log \left(x\right)\right)\right]}^{m+1}}{m+1}+c$