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`

Explanation

To prove the equation \( 2n I_{n+1} = 2^{-n} + (2n - 1) I_n \), where \( I_n = \int_0^1 \frac{dx}{(1+x^2)^n} \), we will follow these steps:

### Step 1: Write the expression for \( I_n \)
We start with the definition of \( I_n \):
\[ I_n = \int_0^1 \frac{dx}{(1+x^2)^n} \]

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