Question

For the following exercises, determine whether the statement is true or false. Justify your answer with a proof or a counterexample\(\int \frac{1}{x^{4}+1} d x\) cannot be integrated using partial fractions.

Short answer

False; partial fractions can't be directly used due to the irreducible nature of \( x^4 + 1 \) over real numbers.

Step-by-step solution

Identify the Need for Partial Fractions

We start by analyzing the integrand \( \frac{1}{x^4+1} \). For partial fraction decomposition, the denominator should be factored into simpler, irreducible polynomials.

Factor the Denominator

To use partial fractions, the denominator \( x^4 + 1 \) needs to be factored. We consider possible factorizations. We can use the identity \( x^4 + 1 = (x^2 + \sqrt{2}x + 1)(x^2 - \sqrt{2}x + 1) \) but after further simplification with complex numbers, unwanted results occur, so we check if it's factored over real numbers as \((x^2 + x\sqrt{2} + 1)(x^2 - x\sqrt{2} + 1)\) which is incorrect. Therefore, \( x^4 + 1 \) is not factorable into real polynomials of lower degree.

Consider Complex Roots

An alternative method is to factor using complex numbers. Solving \( x^4 = -1 \) we get \( x = e^{i(\frac{\pi}{4} + k\pi)} \), which provides four complex roots. Thus, we can express the polynomial as a product of linear factors \((x-e^{i\theta})(x-e^{-i\theta})(x+e^{i\theta})(x+e^{-i\theta})\) where each factor involves complex numbers. This does not allow straightforward partial fraction decomposition for real integrals.

Conclusion on Partial Fractions Feasibility

Since the denominator \( x^4+1 \) does not factor neatly into a product of lower-degree polynomials with real coefficients, and complex factorization does not help with standard partial fraction decomposition in the realm of real numbers, partial fractions cannot be directly applied.

Key concepts

Complex Roots

When dealing with complex roots, you encounter solutions to polynomial equations that are not real numbers. These roots arise when you solve equations like \(x^4 = -1\). The solutions to these types of equations involve complex numbers, which have a real part and an imaginary part. To express these solutions, you often use Euler's formula, linking trigonometry and exponential functions.When you find the complex roots of a polynomial like \(x^4 + 1\), you get expressions such as \(x = e^{i(\frac{\pi}{4} + k\pi)}\), where \(k\) is an integer. Here, \(e^{i\theta}\) is the exponential form of a complex number, and using this formula, we discern four distinct complex roots corresponding to different values of \(k\). These roots are symmetric about the origin in the complex plane, often making it challenging to apply certain algebraic methods directly, especially when working within the real number system.

Factorization

The factorization of polynomials is crucial in simplifying expressions and solving equations. In the context of partial fractions, having smaller, simpler pieces makes integration manageable. For \(x^4 + 1\), one aim could be to express it as a product of lower-degree polynomials.Using algebraic identities can occasionally guide factorization, but for polynomials with higher degrees, especially those without straightforward roots, factorization can become complex. Often, factoring over the reals is impossible if complex coefficients are not permitted, as is the case here. Attempting to factorize \(x^4 + 1\) over the real numbers doesn't yield usable factors for partial fraction decomposition, pushing us to consider complex numbers.

Polynomial Integration

Polynomial integration involves finding the antiderivative of polynomial expressions. If a polynomial can be broken down into simpler fractions, integration becomes much simpler. Each term in a partial fraction setup often corresponds to a basic integral form we can solve.However, to use partial fractions, the polynomial needs to be split into irreducible factors over the reals, which isn’t feasible for \(x^4 + 1\). Without this factorization into simpler terms, standard techniques for polynomial integration can't apply, meaning alternative methods or numerical approximations might be necessary for the integration of more complex polynomials.

Real Coefficients

Concerning polynomials in real algebra, real coefficients are crucial as they imply that the numbers leading terms in polynomials are real, and typically we aim to keep it that way when factorizing.A polynomial with real coefficients means that its non-leading terms, and solutions, when wholly factorized, should also be expressible in terms that reflect real number parts. When complex roots are involved, ensuring that polynomial decomposition in terms producing real coefficients is practically impossible, as was demonstrated with \(x^4 + 1\). Therefore, for uses like partial fraction decomposition, focusing solely on real coefficients can limit the applicability when dealing with higher-degree polynomials.