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mjolnir80
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Homework Statement
find the indefinate integral of 1+x2/(1+x4)
Homework Equations
The Attempt at a Solution
im having trouble trying to figure what to use to substitute into this equation
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.
The purpose of finding the integral of 1+x^2/(1+x^4) is to determine the area under the curve of the given function. This can be useful in many applications, such as calculating work done in physics or finding the total cost of production in economics.
The step-by-step solution for finding the integral of 1+x^2/(1+x^4) involves using substitution and the properties of integrals. First, substitute u = x^2 + 1 to simplify the integral to 1/u(1+u^2). Then, use partial fractions to break down the integrand into simpler fractions. Finally, use the inverse substitution to find the final solution.
Yes, the method used to find the integral of 1+x^2/(1+x^4) is known as substitution. This involves substituting a variable to simplify the integral and then using the inverse substitution to find the final solution. It is a commonly used method in calculus to solve various types of integrals.
No, the integral of 1+x^2/(1+x^4) cannot be solved without using substitution. Other methods, such as integration by parts or trigonometric substitutions, are not applicable to this particular integral. Therefore, substitution is the most efficient and effective method for solving this integral.
The final result of the integral of 1+x^2/(1+x^4) is 1/2ln(1+x^4) + 1/4arctan(x^2) + C, where C is the constant of integration. This can be verified by differentiating the solution to get back the original integrand. It is important to include the constant of integration as it accounts for all possible solutions to the integral.