Question

$\int \frac{1}{\log \left(x^x\right)(1+\log x)}dx=$

• A. $\log |1+\log x|+c$
• B. $\log |\frac{1+\log x}{\log x}|+c$
• C. $\log |\frac{\log x}{1+\log x}|+c$
• D. $\log |\frac{1+\log x}{x\log x}|+c$

Answer 1

Answer: (C)

$\log |\frac{\log x}{1+\log x}|+c$

$logx=t$

${}^1{/}_{x}dx=dt$

$\int \frac{1}{xlogx(1+logx)}dx$

$\int \frac{dt}{t(1+t)}=\int (\frac{1}{t}-\frac{1}{1+t})dt$

$\int \frac{1}{t}dt-\int \frac{1}{(1+t)}dt$

$=log\left(t\right)-log(1+t)+c$

$=log\left(\frac{t}{1+t}\right)+c$

$\int \frac{1}{\left(log{x}^x\right)(1+logx)}dt=log\left(\frac{logx}{1+logx}\right)+c$