`int_(0)^((2)/(3))(dx)/(4+9x^(2))` बराबर है :

Options

A. `(pi)/(6)`
B. `(pi)/(12)`
C. `(pi)/(24)`
D. `(pi)/(4)`

Answer

C

Explanation

`int_(0)^(2//3)(1)/(4+9x^(2))dx`
`=(1)/(9) int_(0)^(2//3)(1)/((4)/(9)+x^(2))dx`
`=(1)/(9)int_(0)^(2//3)(1)/(((2)/(3))^(2)+x^(2))dx`
`=(1)/(9).(1)/(2//3)[tan^(-1)((x)/(2//3))]_(0)^(2//3)`
`=(1)/(6)[tan^(-1)((3x)/(2))]_(0)^(2//3)`
`=(1)/(6)[tan^(-1)((3)/(2).(2)/(3))-tan^(-1)0]`
`=(1)/(6)(tan^(-1)1-0)=(1)/(6).(pi)/(4)=(pi)/(24)`