It is given that I(n) = Int[x^n/(x^4+1) dx], x = 0 to x =
1
I(n+4) + I(n)
=>
Int[x^(n+4)/(x^4+1) dx], x = 0 to x = 1 + Int[x^n/(x^4+1) dx], x = 0 to x =
1
=> Int[x^(n+4)/(x^4+1) dx] + Int[x^n/(x^4+1) dx],
x = 0 to x = 1
=> Int[(x^(n+4)/(x^4+1) +
x^n/(x^4+1)) dx], x = 0 to x = 1
=> Int[(x^(n+4) +
x^n)/(x^4+1) dx], x = 0 to x = 1
=> Int[(x^n(x^4 +
1)/(x^4+1) dx], x = 0 to x = 1
=> Int[x^n dx], x = 0
to x = 1
=> [x^(n + 1)]/(n + 1), x = 0 to x =
1
=> 1^(n + 1)/(n + 1) - 0^(n + 1)/(n +
1)
=>
1/(n+1)
This proves
I(n+4)+I(n)=1/(n+1)
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