91Ó°ÊÓ

Polynomial factorization is an essential concept that involves breaking down a polynomial expression into a product of simpler polynomials, often to make calculations or integrations easier. In our exercise, the integrand has a polynomial in the denominator: \( (x^2 + 4)(3-x) \). Factorizing or recognizing these polynomials' form allows us to use techniques like partial fraction decomposition effectively.

To factorize polynomials, you should:

• Identify common factors (i.e., terms that can be taken out of each component).
• Factor any quadratic or higher-degree polynomials using techniques such as grouping, using the quadratic formula, or recognizing special product forms (e.g., difference of squares).
• Express the polynomial as a product of simpler factors, each of which ideally cannot be factored further.

In our example, the expression doesn't need further factorization since it is already split into manageable factors: \( x^2 + 4 \) and \( 3-x \). These are simpler polynomials that make partial fraction decomposition viable.

Partial fraction decomposition is a powerful method used primarily to simplify mathematical expressions, particularly when dealing with rational functions. This approach decomposes a complex fraction into a sum of simpler fractions, which are often more straightforward to integrate or differentiate.

In this exercise, we started with the expression \( \frac{3x + 4}{(x^2 + 4)(3-x)} \). By recognizing the distinct polynomial factors in the denominator, it is possible to express it as a sum of simpler fractions:

• \( \frac{Ax + B}{x^2 + 4} \)
• \( \frac{C}{3-x} \)

Here, \( A \), \( B \), and \( C \) are constants that we need to determine. This is achieved by clearing the denominators and manipulating the equation to equate coefficients on both sides. Through systematic solving of the equations obtained from equating coefficients, we identify these constants and simplify the fraction.

Once decomposed, each term can be tackled individually, making complex integrations much more manageable, as simplifying complex expressions into linear or lesser-degree terms often allows straightforward application of basic integration techniques.

Integral calculus involves finding the integral of a function, which is essentially the reverse process of differentiation. Integrals can help determine areas under curves, among other applications, and they come in two main types: indefinite and definite integrals.

In the given exercise, we dealt with an indefinite integral \( \int \frac{3x+4}{(x^2+4)(3-x)} \, dx \). After performing partial fraction decomposition, the task was to integrate each simplified term separately:
We have:

• \( \int \frac{\frac{7}{2} x}{x^2+4} \, dx \)
• \( \int \frac{4}{x^2+4} \, dx \)

By recognizing integral forms and applying substitution methods where necessary, we simplify the integration process. For example:

• Using substitutions like \( u = x^2 + 4 \) which helps integrate polynomials in fractions with different bases.
• Employ known integral formulas such as \( \int \frac{1}{x^2+a^2} \, dx = \frac{1}{a} \arctan\left(\frac{x}{a}\right) + C \).

These techniques allow us to arrive at the solution, expressing it as a combination of simpler integrals, which becomes \( \frac{7}{4} \ln|x^2 + 4| + 2 \arctan\left(\frac{x}{2}\right) + C \). Integrals can thus be simplified and solved, even if initially they seem daunting.