91Ó°ÊÓ

Polynomial expansion is a technique used to simplify expressions raised to a power. When you see a polynomial like \((x^2 + 1)^2\), you're looking at a binomial raised to an exponent. A common formula used here is \((a + b)^2 = a^2 + 2ab + b^2\). This allows you to expand the expression efficiently. Applying this to \((x^2 + 1)^2\), we treat \(x^2\) as \(a\) and \(1\) as \(b\), resulting in the expression \(x^4 + 2x^2 + 1\). The expanded polynomial is now easier to work with for integration because each term can be integrated independently.
By expanding the polynomial, we simplify the process of finding integrals. We break it down into easy, manageable parts.

Integration techniques are methods used to find the integral of a function. Once we've expanded our polynomial, we use straightforward techniques for integrating each term. Each term of the polynomial \(x^4 + 2x^2 + 1\) can be integrated separately using basic rules of integration:

• For \(x^n\), the integral is \(\frac{x^{n+1}}{n+1} + C\) where \(neq -1\).
• The integral of a constant, like 1, is simply \(x + C\).

Applying these rules:

• Integrate \(x^4\) to get \(\frac{x^5}{5} + C\).
• Integrate \(2x^2\) to get \(\frac{2x^3}{3} + C\).
• Integrate \(1\) to get \(x + C\).

Each term's constant \(C\) represents potential horizontal shifts in the integral, but we often summarize with a single \(C\) at the end.

While the original task involves finding an indefinite integral, it's useful to understand definite integrals too. Definite integrals calculate the accumulated value over an interval \([a, b]\). Say you need to find it from 0 to 1, denoted as \(\int_{0}^{1} (x^2 + 1)^2 \, dx\):
First, integrate the expanded form \(x^4 + 2x^2 + 1\) as we've practiced. Then evaluate \(\left[\frac{x^5}{5} + \frac{2x^3}{3} + x \right]_0^1\). You plug in the upper limit:\[\frac{(1)^5}{5} + \frac{2(1)^3}{3} + 1 = \frac{1}{5} + \frac{2}{3} + 1 = 1.8667\]
And the lower limit (which will be zero in this case because the terms evaluated at \(0\) are zero).
The definite integral provides a specific accumulated value and gives insights into the area under the curve defined by the function from \(a\) to \(b\). This process enhances our understanding of how these functions behave within certain constraints.