Question

If $ \int{\frac{dx}{x\,\log \,x}}=f(x)+ $ constant, then $ f(x) $ is equal to

• A. $ 1/\,\log \,x $
• B. $ \log \,x $
• C. $ \log \,\,\log \,x $
• D. $ x/\,\log \,x $

Answer 1

Answer: (C)

We have $ \int{dx/(x\,\log x)=f(x)+\text{constant}} $ ..(i)
Let $ \log \,x=t $
$ \Rightarrow $ $ \frac{1}{x}\,\,dx=dt $
$ \therefore $ $ \int{\frac{dx}{x\,\,\log \,x}}=\int{\frac{dt}{t}}=\log (t)+constant $
$ =\log \,(\log x)+\text{constant} $
On comparing with Eq. (i), we get $ f(x)=\log \,(\log x) $