Evaluate : `int_(a)^(b)(logx)/(x) dx `.

Options

A. `(1)/(2)[(log b) - (log a )]`
B. `(1)/(2)[(log a)^2 - (log b )^2]`
C. `(1)/(2)[(log b)^2 - (log a )^2]`
D. `[(log b)^2 - (log a )^2]`

Answer

C

Explanation

Let log `x =z` , so `(1)/(x)dx =dz`.
when `x = a, z` =log a and when `x = b,z`= log b
Hence, `oversetbundersetaint(logx)/(x) dx = overset(log b)underset(log a)int z dz = ((z^2)/(2)) overset(log b)underset(log a)int =(1)/(2)[(log b)^2 - (log a )^2]`