`int(logsqrtx)/(3sqrtx)dx` is equal to

Options

A. `(1)/(3)(log sqrtx)^(2)+C`
B. `(2)/(2)(log sqrtx)^(2)+C`
C. `(2)/(3)(logx)^(2)+C`
D. `(1)/(3)(logx)^(2)+C`

Answer

A

Explanation

Let `l=int(logsqrtx)/(3x)dx`
Put`" "sqrtx=t`
`(1)/(sqrtx)dx=2dt`
`therefore" "l=int(2logt)/(3t)dt=(2)/(3)int(logt)/(t)dt`
Again, put `log t = mu rArr (1)/(t)dt= d mu`
`therefore" "l=(2)/(3)int mu d mu=(2)/(3)(mu^(2))/(2)+C`
`=(2)/(3).((logt)^(2))/(2)+C=(1)/(3)(log sqrtx)^(2)+C`