The value of `int_(0)^(4pi)log_(e)|3sinx+3sqrt(3) cos x|dx` then the value of I is equal to

Options

A. `pi log_(e)3`
B. `2 pi log_(e)3`
C. `4pilog_(e)3`
D. `8pilog_(e)3`

Answer

C

Explanation

`I=int_(0)^(4pi)log_(e)|6sin(x+(pi)/3)|dx`
`=int_(0)^(4pi)log_(e) 6 dx+int_(0)^(4pi)|sin(x+(pi)/3)|dx`
`=4pi log_(e) 6+4int_(0)^(pi)log_(e)|sinx|dx`
`=4pi log_(e)6+8int_(0)^(pi//2) log_(e)|sinx|dx`
`=4pi log_(e)6-4log_(e)2`
`=4pi log_(e)3`