The antiderivative of 1/(x^4+1): Part 1

Here’s an innocuous looking integral:

\displaystyle \int \frac{1}{x^4 + 1} dx

This integral arguably has the highest ratio of “really hard to compute” to “really easy to write” of any indefinite integral, since it is merely a rational function without any powers with non-integer exponents, trigonometric functions, exponential functions, or logarithms. Furthermore, the numerator is a constant while the denominator has only two terms. It doesn’t look that hard.

But this integral is really hard to compute. Indeed, in my experience, this integral is often held as the gold standard for Calculus II (or AP Calculus) students who are learning the various techniques of integration. In this series, I will discuss the various methods that have to be employed to find this antiderivative.

I’ll begin this tomorrow. In the meantime, I’ll leave a thought bubble if you’d like to think about how to compute this integral.


3 thoughts on “The antiderivative of 1/(x^4+1): Part 1

  1. It is So long ago I did this sort of stuff !
    I factorised x^4+1, did the partial fractions, and got bored. Then I remembered about my algebra program in which I implemented a lovely algorithm for finding the quadratic factors of any polynomial. The reason I am telling you all this is that x^4+1 was the only polynomial that te algorithm refused to deal with properly.

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