`int_(0)^(pi//2) (dx)/(1+root(3)(tanx))` is equal to

Options

A. `pi/2`
B. 0
C. `pi/4`
D. None of these

Answer

C

Explanation

We have ` int_(0)^(pi//2) (dx)/(1+3sqrt(tanx ))`
` l= int_(0)^(pi//2) (cos^(3//2)x)/(cos^(3//2) + sin^(3//2)x) dx`
`l= int_(0)^(pi//2)(cos^(3//2)(pi/2-x))/(cos^(3//2)(pi/2-x)+sin^(3//2)(pi/2-x))dx`
` 2l=int_(0)^(pi//2) (cos^(3//2) x +sin^(3//2)x)/(cos^(3//2)x + sin^(3//2)x)dx`
` int_(0)^(pi//2) dx = pi/2`
` Rightarrow l= pi/4`