If `int_(0)^(k)(dx)/(2+18x^(2))=(pi)/(24)`, then the value of k is

Options

A. 3
B. 4
C. `(1)/(3)`
D. `(1)/(4)`

Answer

C

Explanation

We have,
`int_(0)^(k)(dx)/(2+18x^(2))=(pi)/(24)`
`implies(1)/(18)int_(0)^(k)(dx)/(((1)/(3))^(2)+x^(2))=(pi)/(24)`
`implies(1)/(18)xx(1)/((1)/(3))[tan^(-1)3x]_(0)^(k)=(pi)/(24)`
`implies[tan^(-1)3k-10^(-1)0]=(pi)/(4)`
`impliestan^(-1)3k=(pi)/(4)`
`implies3k="tan"(pi)/(4)`
`implies3k=1impliesk=(1)/(3)`