The value of the integral `int_0^(1/2)(1+sqrt(3))/(((x+1)^2(1-x)^6)^(1/4))dx` is ______.

Answer

2

Explanation

`I=int_(0)^(1//2)(1+sqrt(3))/(((x+1)^(2)(1-x)^(6))^(1//4))dx`
`=int_(0)^(1//2) ((1+sqrt(3))dx)/((1+x)^(2)[((1-x)^(6))/((1+x)^(6))]^(1//4)`
Put `(1-x)/(1+x)=t`
`implies(-2dx)/((1+x)^(2))=dt`
`:.I=int_(1)^(1//3)((1+sqrt(3))dt)/(-2t^(3//2))=(-(1+sqrt(3)))/2xx|(-2)/(sqrt(t))|_(1)^(1//3)`
`=(1+sqrt(3))(sqrt(3)-1)=2`