What is the indefinate integral of 1/(x^4+1) ?

Answer 1

Well, you have to use partial fraction for this integral. I'm going to give you the links to the solutions. This first link show you the partial fraction decomposition of the function

http://www.calc101.com/webMathematica/partial-fractions.jsp#topdoit

The second link gives you the answer without the steps, but after the partial fraction decomposition you should see how the answer was obtained.
http://www.calc101.com/webMathematica/integrals.jsp#topdoit

Answer 2

Method suggested by NBL is not suitable for manual working. Perhaps, it may be ok for computer programming which mathematica uses.
Here, in India, we teach the students the following method which is the simplest in terms of number of steps. But I, for one, had been an advocate of the method suggested by steiner 1, even though it is lenghier and more cumbersome. The reason is that I consider it as more precise and all steps are mathematically perfect in that route. The method which I have used below has one defective step of division by x^2 not permitted unless x is given to be non-zero and results in a defective answer which many books of calculus allow. Actually, domain of the function includes the values of x = 0. The function is defined and continuous at x = 0, whereas the answer of integration performed hereunder is not defined at x = 0. In view of its simplicity, I have shown steps of the method we teach. If you need any further clarification on what I have stated, let me know. My e-mail is open.

∫ 1 / (x^4+1) dx
= (1/2) ∫[ [(x^2 + 1) - (x^2 - 1)] / (x^4+1) ] dx
= (1/2) ∫ [(x^2 + 1) / (x^4 + 1)] dx - (1/2) ∫ [(x^2 - 1) / (x^4 + 1)] dx
= I + I' ... ... ( 1 )

I = (1/2) ∫ [(x^2 + 1) / (x^4 + 1)] dx
= (1/2) ∫ [(1 + 1/x^2) / (x^2 + 1/x^2)]dx
Let x - 1/x = u => (1 + 1/x^2) dx = du
and x^2 + 1/x^2 = (x - 1/x)^2 + 2
I = (1/2) ∫ du / (1 + u^2) = (1/2) arctan u + c
I = (1/2) arctan ( x - 1/x) + c

Similarly, work out I' taking x + 1/x = v, (1 - 1/x^2) dx and
x^2 + 1/x^2 = (x + 1/x)^2 - 2 = v^2 - 2.
Put the values of I and I' in eqn. ( 1 ) to get the answer.
Let me know if you still have problem solving it.

Answer 3

This is an exercise in partial fractions.
The hardest step, IMHO is to factor the
denominator:
x^4 + 1 = (x² + √2 x + 1)(x² - √2x + 1).
Now use partial fractions to get the answer.