Integral of 1/(1+x^4)^1/2?

Question

How would I even start this one? I'm thinking it's a Fundamental Theorem of Calculus problem, but any help would be appreciated.

Answer 1

I agree with the person above me; I don't think this can be integrated using methods learned in a calculus class (or even in an undergraduate analysis class).

You can find its antiderivative at http://integrals.wolfram.com/index.jsp . It is complicated and uses obscure functions. (And one of the functions it uses is "cheating" because that function uses an integral in its definition.)

If you assume 0 <= x <= pi/2, then you can use the trigonometric substitution x = √(tan θ) to reduce the integral to finding ∫(sec θ / (2√(tan θ)) dθ, but as this isn't possible to integrate using methods from a calculus class either, this doesn't seem very useful. So I won't bother to show my work here.

Answer 2

Sorry, this is an elliptic integral.
There is no antiderivative in terms of elementary
functions, so the fundamental theorem of calculus
won't work here.
In general, when you have to integrate the
square root of a cubic or quartic polynomial, you
have an elliptic integral.