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If In=int0^1(dx)/((1+x^2)^n); n in N , ...

If `I_n=int_0^1(dx)/((1+x^2)^n); n in N ,` then prove that `2nI_(n+1)=2^(-n)+(2n-1)I_n`

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`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)`
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