The problem involves finding the indefinite integral of the expression (1 + x²)/(1 + x⁴), which is situated within the context of integral calculus.
The discussion is ongoing, with various approaches being explored. Some participants have provided insights into the factorization of x⁴ + 1, while others express skepticism about its reducibility over the reals. There is no explicit consensus on the best method to proceed.
Participants note that x⁴ + 1 is irreducible over the rationals but can be factored into quadratics with real coefficients. There is also mention of the roots being complex, which may influence the approach to the integral.
I'd like to see that factorization. I've been under the impression for a long time that x^4 + 1 was irreducible over the reals.Dick said:I would do it by partial fractions. You can factor (x^4+1) into two real quadratics. The coefficients may not all be rational, but that's ok. Can you do that?
Mark44 said:I'd like to see that factorization. I've been under the impression for a long time that x^4 + 1 was irreducible over the reals.
If we allow complex numbers, the fourth roots of -1 (solutions of x^4 = -1) are
cos(pi/4) + i sin(pi/4)
cos(3pi/4) + i sin(3pi/4)
cos(5pi/4) + i sin(5pi/4)
cos(7pi/4) + i sin(7pi/4)
none of which are real.
OK, let me rephrase: x^4 + 1 can't be factored into linear factors with real coefficients.Dick said:Ok. x^4+1=(x^2+sqrt(2)x+1)(x^2-sqrt(2)+1). x^4+1 is irreducible over the RATIONALS, not over the REALS. Your roots group into two pairs of complex conjugates (of course). Use them to construct the quadratics.