Question

Evaluate : `(i) int_(0)^(1//2)(dx)/(sqrt(1-x))` `(ii) int_(0)^(1)((1-x)/(1+x))dx`

Answer 1

`(i) int_(0)^(1//2)(dx)/(sqrt(1-x))=int_(0)^(1//2)(1-x)^(-1//2)dx=[(2sqrt(1-x))/(-1)]_(0)^(1//2)=(2-sqrt(2))`.
`(ii) int_(0)^(1)((1-x)/(1+x))dx=int_(0)^(1)(-1+(2)/(x+1))dx` [on dividing `(-x+1)` by `(x+1)`]
`=[-x+2log|x+1|]_(0)^(1)=[(2log2)-1]`.