91Ó°ÊÓ

Polynomial expansion is a fundamental technique in algebra that you use to express a power of a binomial as a polynomial. In the given problem, you need to expand \((x+1)^2\). This is achieved by multiplying \((x + 1)\) by itself: \[(x+1) \times (x+1) = x^2 + 2x + 1\]. This expansion results in three distinct terms, each representing a different power of \(x\). The next step involves distributing \(x^3\) over each term of the expanded polynomial, thus allowing you to create individual terms that are simple enough to integrate later.

• Step 1: Multiply \(x^3\) with \(x^2\), \(2x\), and \(1\) to get \(x^5 + 2x^4 + x^3\).

Understanding polynomial expansion helps simplify many calculus problems by breaking them down into easier, manageable parts.

The power rule is a basic and powerful tool in calculus for finding the derivative and integral of terms raised to a power. In integration, it allows you to integrate terms of the form \(x^n\). The power rule states that for any real number \(n\), except \(n = -1\), the integral of\(x^n\) is \((x^{n+1}/(n+1)) + C\), where \(C\) is the constant of integration.

The power rule makes it simple to integrate polynomial functions by dealing with each term individually. In this exercise:

• For \(\int x^5 \, dx\): apply the rule (where \(n = 5\)), to get \(\frac{x^6}{6}\).
• For \(\int 2x^4 \, dx\): use \(2\) as a constant multiplier, so it remains, giving \(\frac{2x^5}{5}\).
• Finally for \(\int x^3 \, dx\), applying the rule yields \(\frac{x^4}{4}\).

The power rule thus provides a systematic way to find antiderivatives for polynomial terms, streamlining the process.

Integrating polynomials is a building block skill in calculus, where you find the antiderivative of polynomial expressions. We use the power rule here, as it provides a straightforward way to integrate each term individually.

In this problem, once the polynomial \((x+1)^2 x^3\) is expanded using polynomial expansion, it results in simpler terms, \(x^5, 2x^4,\) and \(x^3\). Each of these terms can be integrated independently using their respective power rules. By integrating each separate term:

• Integrating \(x^5\) results in \(\frac{x^6}{6}\).
• Integrating \(2x^4\) results in \(\frac{2x^5}{5}\).
• Integrating \(x^3\) results in \(\frac{x^4}{4}\).

After integrating the terms, you combine them, adding a constant \(C\) at the end since integration can include any constant term. This approach makes handling polynomial integrals easier by simplifying the terms and applying known rules, culminating in the solution: \(\frac{x^6}{6} + \frac{2x^5}{5} + \frac{x^4}{4} + C\). Understanding each step ensures a thorough grasp of performing basic polynomial integrals.