91Ó°ÊÓ

Polynomial division is a powerful technique used to simplify complex rational functions before they are integrated. Here's how it works: when you have a polynomial in the numerator and another in the denominator, like in the expression \(\frac{x^{2} + x + 1}{x^{2} - 1}\), your goal is to simplify it. By dividing the numerator by the denominator, you attempt to express it as a sum of simpler fractions.
For our example, dividing \(x^2 + x + 1\) by \(x^2 - 1\), the quotient is 1 with a remainder of \(x\). This means the original fraction can be expressed as

• 1, plus
• \(\frac{x}{x^2 - 1}\)
• plus \(\frac{1}{x^2 - 1}\).

This decomposition is easier to integrate. Polynomial division is especially helpful when the degree of the numerator is greater than or equal to the denominator.

Partial fraction decomposition involves breaking down a complex rational expression into simpler fractions that are easier to integrate.
When dealing with the fraction \(\frac{1}{x^{2} - 1}\), notice that it can be factored into \((x - 1)(x + 1)\). The decomposition aims to express this fraction as:

• \(\frac{A}{x - 1}\)
• plus \(\frac{B}{x + 1}\)

where \(A\) and \(B\) are constants to be determined.After setting the expressions equal and solving for \(A\) and \(B\), you'll find that \(A = \frac{1}{2}\) and \(B = -\frac{1}{2}\). This yields the expression:

• \(\frac{1}{2}\frac{1}{x - 1}\) minus
• \(\frac{1}{2}\frac{1}{x + 1}\)

This form allows you to integrate each term separately with more ease. Partial fraction decomposition is particularly useful when the integrand can be expressed as a polynomial over a polynomial.

Logarithmic integration comes into play when integrating fractions where the denominator can be expressed as part of a natural log function's derivative. Here, terms like \(\int \frac{1}{u} \, du = \ln|u| + C\) are involved.
In our exercise, after substitution, the integral \(\int \frac{x}{x^2 - 1} \, dx\) simplifies to \(\ln|x^2 - 1|\). This was achieved by using the substitution method to set \(u = x^2 - 1\), leading to \(\int \frac{1}{u} \, du\).
Similarly, derivatives of natural logs appear in the partial fraction decomposition when we find:

• \(\int \frac{1}{x - 1} \, dx = \ln|x - 1| + C\)
• \(\int -\frac{1}{x + 1} \, dx = -\ln|x + 1| + C\)

Logarithmic integration is helpful in handling integrals of the form \(\frac{x}{a^2 + x^2}\), revealing the relationship between derivatives and logarithmic functions.

The substitution method, also known as \(u\)-substitution, is a technique where we change variables to simplify integration. In our case, it is used to solve \(\int \frac{x}{x^2 - 1} \, dx\).
Here, we made the substitution \(u = x^2 - 1\), which simplifies the integrand, allowing us to write the differential \(du = 2x \, dx\), or equivalently \(\frac{1}{2} \, du = x \, dx\). By replacing \(x \, dx\) with \(\frac{1}{2} \, du\), the integral turns into:

• \(\frac{1}{2} \int \frac{1}{u} \, du\),
• which then simplifies to \(\frac{1}{2} \ln|u| + C\).

This technique is highly valuable when an integrand contains a function and its derivative, turning a complex function into a simple integral.