91Ó°ÊÓ

Polynomial long division is a technique used to simplify complex rational expressions. In our exercise, we need to divide the polynomial in the numerator \(4x^3 - 1\) by the polynomial in the denominator \(2x - 1\). This makes it easier to integrate.

Just like long division with numbers, polynomial long division involves several steps:

• Divide the leading term of the numerator by the leading term of the denominator.
• Multiply the entire divisor by this quotient and subtract the result from the original polynomial.
• Repeat the process with the new polynomial formed until the remaining polynomial is of lower degree than the divisor.

For our exercise, we start by dividing \(4x^3\) by \(2x\), which gives us \(2x^2\). Then, we multiply \(2x - 1\) by \(2x^2\) to get \(4x^3 - 2x^2\). We subtract this from \(4x^3 - 1\), which leaves us with \(2x^2 - 1\).

We repeat the process with \(2x^2\), giving us a new term in our quotient, and continue until the degree of the remaining term is less than the degree of the denominator. This step-by-step process simplifies the rational expression for easier integration.

Once the function is simplified using polynomial long division, we can apply various integration techniques to solve the integral. Here, integration refers to finding the antiderivative of the function.

After dividing, we may need to use different techniques such as:

• Basic antiderivatives: Use known basic integration rules, e.g., \( \frac{d}{dx} x^n = \frac{x^{(n+1)}}{n+1} \) for power functions.
• Substitution: Replace a part of the integral with a new variable to simplify.
• Partial fractions: Decompose complex rational expressions into simpler fractions.

In our specific exercise, the resulting polynomial after division will be easier to integrate term by term. Typically, each term will fit into one of the basic forms we know how to integrate. For example, the integral of \(x^n\) is \( \frac{x^{n+1}}{n+1} \), and constants integrate to \(c \cdot x \). This makes the process straightforward once the polynomial is divided.

Definite integrals are different from indefinite integrals because they evaluate the integral within given limits, from \(a\) to \(b\). The result represents the accumulated area under the curve between these points.

For our exercise, we calculate the definite integral from \(x = 1 \) to \(x = 5\). After integrating the simplified function, you'll need to evaluate the antiderivative at the upper limit and subtract the value when evaluated at the lower limit.

This process can be summarized into a few steps:

• Find the antiderivative, which is the indefinite integral without bounds.
• Plug in the upper limit \(b\) into this antiderivative function.
• Plug in the lower limit \(a\) into the same function.
• Subtract the result of the lower limit evaluation from the upper limit evaluation.

In our example, let's say the simplified integral yields an antiderivative \(F(x)\). Then, we find \(F(5) - F(1)\). The resulting value gives the definite integral of the function in the specified range.