# How do you use partial fraction decomposition to decompose the fraction to integrate #1/((x^4) +1)#?

##### 1 Answer

The first (and main) challenge is to factor

#### Explanation:

But fear not, haven't we learned (been told) that every polynomial with real coefficients can be factored into a product of constant, linear and quadratic polynomials with real coefficients? (If we haven't been told that, we should have been.)

We also have need told that every degree n polynomial has n zeros (counting multiplicity). What are the zeros of

They are the solutions to

so they solve

and

or

So we need the square roots of

Well, sort of. We know that if

So all we need is just one of the 4 imaginary solutions and we can get the others.

Unless you know something about the geometry of complex numbers (the complex plane) or some version of De Moivre's Theorem, or at least have had a teacher mention that

But

and the conjugate of that last one:

Let's use the usual notation

The conjugate pairs theorem works as it does because

Lets multiply:

Multiply the other pair of factors:

Finally, just to convince yourself it worked, multiply it out to verify that

Now, back to the integral:

For the partial fraction decomposition, you need

Have fun!