Question

$ \int\limits_1^e \log \, x \, dx = $

• A. $1$
• B. $e - 1$
• C. $e + 1$
• D. $0$

Answer 1

Answer: (A)

We have, $ \int\limits_1^e \log \, x \, dx = $
$=\left[\log x \int\limits 1 dx \right]^{e}_{1} - \int\limits_1^e \left( \frac{d}{dx} \log \, x \, \int\limits 1 \, dx \right) dx $
$=\left[\log\, x . x \right]^{e}_{1} - \int\limits_1^e \frac{1}{x} x \, dx $
$ = [e \, \log \, e - 1 \, \log (1) ] - \int\limits_1^e 1 \, dx$
$ = e - [x]^e_1 =e - [e - 1] = 1$