Thursday, November 19, 2015

Verify if I(n+4)+I(n)=1/(n+1), if I(n)=Integral of x^n/(x^4+1)?limits of integration ar x=0 and x=1

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)

No comments:

Post a Comment

What accomplishments did Bill Clinton have as president?

Of course, Bill Clinton's presidency will be most clearly remembered for the fact that he was only the second president ever...