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

The first (and main) challenge is to factor

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

The conjugate pairs theorem works as it does because

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

Now, back to the integral:

Have fun!

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To decompose the fraction ( \frac{1}{{x^4 + 1}} ) using partial fraction decomposition, you would first factor the denominator as ( (x^2 + 1)(x^2 - 1) ). Since ( x^2 - 1 ) can be further factored as ( (x + 1)(x - 1) ), the denominator factors into ( (x + 1)(x - 1)(x^2 + 1) ).

Now, the partial fraction decomposition of ( \frac{1}{{x^4 + 1}} ) will have the form ( \frac{A}{{x + 1}} + \frac{B}{{x - 1}} + \frac{Cx + D}{{x^2 + 1}} ).

To find the values of ( A ), ( B ), ( C ), and ( D ), you would multiply both sides of the equation by the denominator ( (x^4 + 1) ), then equate coefficients of like terms on both sides of the equation.

Once you find the values of ( A ), ( B ), ( C ), and ( D ), you can integrate each term separately. The integral of ( \frac{A}{{x + 1}} ) will be ( A \ln{|x + 1|} ), the integral of ( \frac{B}{{x - 1}} ) will be ( B \ln{|x - 1|} ), and the integral of ( \frac{Cx + D}{{x^2 + 1}} ) will involve an arctangent function.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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