`IfI_n=int_0^1(dx)/((1+x^2)^n),w h e r en in N ,` which of the following statements hold good? `2nI_(n+1)=2^(-n)+(2n-1)I_n` `I_2=pi/8+1/4` (c) `I_2=pi/8-1/4` `I_3=(3pi)/(32)+1/4`

Explanation

`I_(n)=int_(0)^(1)(dx)/((1+x^(2))^(n))`
`=|1/((1+x^(2))^(n)) . X |_(0)^(1)-int_(0)^(1)n(1+x^(2))^(-n-1)2x.xdx`
`=1/(2^(n))+n int_(0)^(1) (2x^(2))/((1+x^(2))^(n+1))dx`
`=1/(2^(n))+2n int_(0)^(1)(dx)/((1+x^(2))^(n))-2n int_(0)^(1)(dx)/((1+x^(2))^(n+1))`
`=1/(2^(n))+2nI_(n)-2nI_(n+1)`
or `(2n-1)I_(n)+1/(2^(n))=2nI_(n+1)`