- #1
totentanz
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Can anyone help me solving x2+1/x4+1...thanks
Caramon said:Did you use partial fractions to get that result? I think he may be referring to the integral of ((x^2 + 1)/(x^4 + 1))dx
Caramon said:Did you use partial fractions to get that result? I think he may be referring to:
[tex]
\int{\frac{{x^2}+1}{{x^4}+1}dx}
[/tex]
ThomasEgense said:This is the solution, however getting there is not clear :)
http://integrals.wolfram.com/index.jsp?expr=(x*x+1)/(x*x*x*x+1)&
HallsofIvy said:[itex]x^4+ 1= 0[/itex] has no real roots but we can write [itex]x^4= -1[/itex] so that [itex]x^2= \pm i[/itex] and then get [itex]x= \sqrt{2}{2}(1+ i)[/itex], [itex]x= \sqrt{2}{2}(1- i)[/itex], [itex]x= \sqrt{2}{2}(-1+ i)[/itex], and [itex]x= \sqrt{2}{2}(-1- i)[/itex] as the four roots. We can pair those by cojugates:
[tex]\left(x- \frac{\sqrt{2}}{2}(1+ i)\right)\left(x- \frac{\sqrt{2}}{2}(1- i)\right)[/tex][tex]= \left((x- \frac{\sqrt{2}}{2})- i\frac{\sqrt{2}}{2}\right)\left((x- \frac{\sqrt{2}}{2})+ i\frac{\sqrt{2}}{2}\right)[/tex][tex]= (x- \sqrt{2}{2})^2+ \frac{1}{2}= x^2- \sqrt{2}x+ 1[/tex]
and
[tex]\left(x- \frac{\sqrt{2}}{2}(-1+ i)\right)\left(x- \frac{\sqrt{2}}{2}(-1- i)\right)[/tex][tex]= x^2+ \sqrt{2}x+ 1[/tex]
That is, the denominator [itex]x^4+ 1[/itex] factors as
[tex](x^2+ \sqrt{2}x+ 1)(x^2- \sqrt{2}x+ 1)[/tex]
and you can use partial fractions with those.
dextercioby said:Back in my high school days I knew tricks for these. For an anti-derivative sought in a subset of R excluding 0, we can make the following tricks:
[tex] \int \frac{x^2 +1}{x^4 +1} \, dx = \int \frac{1+\frac{1}{x^2}}{x^2 + \frac{1}{x^2}} \, dx = \int \frac{d\left(x-\frac{1}{x}\right)}{\left(x-\frac{1}{x}\right)^2 + \left(\sqrt{2}\right)^2} = ... [/tex]
HallsofIvy said:You have been given a number of different suggestions (dextercioby's making the problem almost trivial) but keep saying you cannot do it- and showing no work at all. Have you tried what dextercioby suggested? What did you do and where did you run into trouble.
totentanz said:I am working on solving it all week,and I can not find a complete answer except what Mr.dextercioby show...and I already solve this "normalization of the wavefunction",thank you for your care
An integral is a mathematical concept that represents the area under a curve in a graph. It is also known as the antiderivative of a function.
Solving integrals is important in many areas of science and engineering, as it allows us to find the total quantity or value of a continuous function over a given interval. It also helps in understanding the behavior of a function.
There are several methods for solving integrals, including substitution, integration by parts, trigonometric substitution, partial fractions, and numerical integration. The choice of method depends on the complexity and form of the integral.
If you are struggling to solve an integral, it is important to review the fundamental concepts of calculus and try different methods. You can also seek help from a tutor or consult online resources and textbooks for practice problems and step-by-step solutions.
You can check your solution by differentiating it and seeing if it matches the original function. You can also use online integral calculators or ask a colleague or tutor to verify your solution.