Evaluate `int_(0)^(pi/2)(tanxdx)/(1+m^(2)tan^(2)x)`

Explanation

Let `I=int_(0)^(pi//2)(tan xdx)/(1+m^(2)tan^(2)x)`
`=int_(0)^(pi//2)((sinx)/(cosx))/(1+m^(2) . (sin^(2) x)/(cos^(2)x)) dx`
`= int_(0)^(pi//2)(sinx cos x dx)/(1-sin^(2)x+m^(2)sin^(2)x) dx`
`=int_(0)^(pi//2)(sinx cosx)/(1+(m^(2)-1)sin^(2)x)`
Put `sin^(2)x=t`
`implies2sin x cos x dx=dt`
when `xto0`, then `t to 0`
and `xto pi/2,` then `t to 1`
`:. I=1/2 int_(0)^(1)(dt)/(1+(m^(2)-1)t)`
`=1/2[1/(m^(2)-1)log(1+(m^(2)-1)t)]_(0)^(1)`
`=1/(2(m^(2)-1))[log(1+(m^(2)-1))-log1]`
`=(log m^(2))/(2(m^(2)-1))`
`=(2log|m|)/(2(m^(2)-1))`
`=(log|m|)/(m^(2)-1)`