What is `int_(0)^(pi//2) (dx)/(a^(2) cos^(2) x+ b^(2) sin^(2) x)` equal to ?

Options

A. `2ab`
B. `2piab`
C. `(pi)/(2ab)`
D. `(pi)/(ab)`

Answer

C

Explanation

Let `I = int_(0)^(pi/2) (dx)/(a^(2) cos^(2) x + b^(2) sin^(2) x)`
`int_(0)^(pi/2) (sec^(2) xdx)/(a^(2) + b^(2) tan^(2)x)`
[divide numerator and denominator by `cos^(2)x`]
Put `tan x = t`
` rArrsec^(2) xdx = dt`
When x =0, then `t = 0` and when `x =pi/2`, then `t = oo`
`:. I = int_(0)^(oo) (dt)/(a^(2) + b^(2) t^(2)) = 1/(b^(2)) int_(0)^(oo) (dt)/((a/b)^(2) t^(2))`
`= 1/(b^(2)) (1)/((a/b)) [tan^(-1) ((bt)/a)]_(0)^(oo)`
`[:' int(dx)/(a^(2) + x^(2)) = 1/a tan^(-1) (x/a) + C]`
`= 1/(ab) [tan^(-1) (oo) - tan^(-1) (0)]`
`= 1/(ab) [pi/2 -0] =pi/(2ab)`