Statement I `int_(pi/6)^(pi/3)1/(1+tan^3x)` is `pi/12` Statement II `int_a^bf(x)dx=int_a^bf(a+b-x)dx`

Options

A. `(pi)/(12)`
B. `(pi)/(4)`
C. `(pi)/(3)`
D. `(pi)/(6)`

Answer

A

Explanation

Let `l=int_(pi//6)^(pi//3)(1)/(1+tan^(3)x)dx`
`=int_(pi//6)^(pi//3)(cos^(3)x)/(sin^(3)x+cos^(3)x)dx" "...(i)`
`impliesl=int_(pi//6)^(pi//3)(sin^(3)x)/(sin^(3)x+cos^(3)x)dx" "...(ii)`
On adding Eqs. (i) and (ii), we get
`2l=int_(pi//6)^(pi//3)1dx=[x]_(pi//6)^(pi//3)=(pi)/(6)impliesl=(pi)/(12)`