Evaluate the following : `int_(0)^(pi//2)(dx)/(a^(2)cos^(2)x+b^(2)sin^(2)x)`

Answer

`(pi)/(2ab)`

Explanation

`I=int_(0)^(pi//2) (dx)/(a^(2)cos^(2)x+b^(2)sin^(2)x)`
`=int_(0)^(pi//2) (sec^(2)x dx)/(a^(2)+b^(2)tan^(2)x)=1/(b^(2)) int_(0)^(pi//2) (sec^(2) xdx)/((a/b)^(2)+tan^(2)x)`
Put `tan x=z` or `sec^(2) x dx=dz`
When `x=0, z=0, xto (pi)/2, zto oo`
`:.I=1/(b^(2)) int_(0)^(oo) (dz)/((a/b)^(2)+z^(2))`
`=1/(b^(2))1/(a/b)|"tan"^(-1)z/(a//b)|_(0)^(oo)`
`=1/(ab)[tan^(-1)oo-tan^(-1)0]`
`=1/(ab) (pi)/2`
`=(pi)/(2ab)`