Evaluate the following :
`int_(0)^(1//2)(dx)/((1-2x^(2))sqrt(1-x^(2)))`

Explanation

Let `I=int_(0)^(1//2)(dx)/((1-2x^(2))sqrt(1-x^(2)))`
Put `x=sin theta" "therefore" "dx= cos theta d theta`
When `x=0, sin theta= 0= sin 0" "therefore theta = 0`
When `x=(1)/(2), sin theta =(1)/(2)=sin.(pi)/(6)" "therefore theta =(pi)/(6)`
`therefore I=int_(0)^(pi//6)(cos theta d theta)/((1-2 sin^(2) theta)sqrt(1-sin^(2)theta))`
`=int_(0)^(pi//6)(cos theta d theta)/(cos 2 theta sqrt(cos^(2) theta))`
`=int_(0)^(pi//6)(cos theta d theta)/(cos 2 theta. cos theta)=int_(0)^(pi//6)(1)/(cos 2 theta)d theta`
`=int_(0)^(pi/6)sec 2theta d theta`
`=(1)/(2)[log|sec 2 theta + tan 2 theta|]_(0)^(pi//6)`
`=(1)/(2)[log(sec.(pi)/(3)+tan.(pi)/(3))-log(sec 0 +tan0)]`
`=(1)/(2)[log(2+sqrt3)-log1]`
`=(1)/(2)log(2+sqrt3)." ..."[because log 1=0]`