91Ó°ÊÓ

The power rule for integration is an essential technique when working with polynomials. It allows you to find the antiderivative of expressions in the form of \( x^n \). This rule can be stated as follows: \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]. Here, \( n \) is any real number, and \( C \) represents the constant of integration.
Understanding this rule significantly simplifies integrating terms with variables raised to a power. When you apply this rule:

• Increase the exponent \( n \) by one; this becomes the new exponent of the variable.
• Divide by this new exponent.
• Don’t forget the constant of integration \( C \).

Let’s take an example: Suppose you want to integrate \( x^2 \). Using the power rule, it transforms to \[ \int x^2 \, dx = \frac{x^{2+1}}{2+1} = \frac{x^3}{3} + C \]
This rule is a cornerstone for calculus and is particularly valuable for polynomial integration.

Polynomial integration involves finding the antiderivative of a polynomial function. A polynomial is typically a sum of terms consisting of variables raised to whole number powers.
To tackle polynomial integration, you integrate each term individually and then sum the results. For example, consider the polynomial \( x^2 + x - 1 \). The integral of this polynomial is determined term-by-term using the power rule.
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