Why is substituting for \int\frac{dx}{x^4+1} so difficult?

In summary, the conversation discusses the integral of 1/x^4+1 and various methods of solving it. The individuals mention trying substitution, partial fractions, and trigonometric substitution but none seem to work. One person suggests a possible solution involving e^(2u)+e^(-2u) and hyperbolic trig functions. Another mentions factoring the denominator and another brings up the idea of the integral being ln|x^4+1|, but this is proven incorrect.
  • #1
Jupiter
46
0
[tex]\int\frac{dx}{x^4+1}[/tex]
What really frustrates me is that I've seen this integral before. I believe it involved some whacky subsitution like [tex]x=e^u[/tex], but no substitution seems to work. Partial fractions just make a mess. Trig subs seem tempting but that 4 screws everything up. Ideas?
 
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  • #2
If you do that whacky substitution, how about then multiplying through to get e^(2u)+e^(-2u) and getting a hyperbolic trig function? Not saying this works.
 
  • #3
[tex]\int\frac{dx}{x^4+1}=\frac{1}{3\sqrt{2}}
\left(2\arctan(1+\sqrt{2}x)-2\arctan(1-\sqrt{2}x)+\right.[/tex]

[tex]\left.\log(\sqrt{2}x+x^2+1)-\log(\sqrt{2}x-x^2-1)\right)[/tex]
 
  • #4
You could factor the denominator!
 
  • #5
isn't the integral of 1/x^4+1 just the ln|x^4+1|? or something like that?
 
  • #6
No, differentiate log(x^4+1) and you'll see why.
 
  • #7
noooo. i mean, the integral of 1/u is the ln|u|+c and in this case, 1/u = (x^4+1)
 
  • #8
In order to do that, you also need to have a full [itex]du[/itex] on your integrand, which would need an [itex]x^3[/itex] term that is not there.
 
  • #9
Also not that if you're correct that the integral of 1/x^2 is log(x^2) and, obviously -1/x as well, so up to a constant 2logx = -1/x?
 

1. What is a substitution integral?

A substitution integral is a method of integration in calculus that involves replacing a variable in an integral with a new variable in order to simplify the integral and make it easier to solve.

2. Why is substitution used in integration?

Substitution is used in integration to make the integration process easier and more efficient. It allows for the integration of complicated functions that would be difficult or impossible to integrate using other methods.

3. How do you choose the substitution variable?

The substitution variable is chosen based on the structure of the integral. The goal is to choose a variable that will result in a simpler integral, often one that involves only a single variable or a known function.

4. What are the steps for solving a substitution integral?

The steps for solving a substitution integral are as follows:1. Identify the substitution variable and rewrite the integral in terms of that variable.2. Take the derivative of the substitution variable and substitute it into the integral.3. Solve the resulting integral.4. Substitute back in the original variable.5. Simplify the integral if possible.

5. Can substitution be used in all integrals?

No, substitution cannot be used in all integrals. It is most useful for integrals that involve complicated functions or expressions. In some cases, it may not be necessary or may not result in a simpler integral.

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