Question

If \[ \int \log x\sqrt{\left(\frac{\log x}{x}\right)^2+\frac1{x^2}}\,dx = \frac{f(x)}{3}\sqrt{1+(\log x)^2}+C \] and \(f(1)=1\), then \(f(e)=\) ?

• A. \(\frac23\)
• B. \(2\)
• C. \(\frac13\)
• D. \(6\)

Hint:

When both \(\log x\) and \(\frac1x\) appear together, try the substitution \(t=\log x\) immediately.

Answer 1

The Correct Option is B