`int_(1)^(x) (log(x^(2)))/(x)` dx is equal to

Options

A. `(log x)^(2)`
B. `(1)/(2) (log x)^(2)`
C. `(log x^(2))/(2) `
D. None of these

Answer

A

Explanation

Let ` I = int_(1)^(x) (log (x^(2)))/(x)dx`
` = 2 int_(1)^(x) log x (1)/(x)dx`
Put `log x = t rArr (1)/(x) dx = dt`
` therefore 2 int_(0)^(log x) t dt = 2[(t^(2))/(2)]_(0)^( log x)`
`= (log x)^(2)`