Which mathematical property is used when integrating a polynomial function?

Which mathematical property is used when integrating a polynomial function?

• A. Linearity of integration
• B. Product rule of integration
• C. Chain rule of integration
• D. Power rule of integration

Answer: (D)

When integrating a polynomial function, the power rule of integration is utilized. This rule states that if you have a term in the form of \(x^n\), where \(n\) is a real number, the integral is given by: \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1) \] This means that when you integrate a polynomial, each term \(ax^n\) in the polynomial can be individually integrated using this rule. The coefficient \(a\) remains the same, and you increase the exponent by one while dividing by the new exponent. This systematic approach allows for straightforward integration of polynomial expressions term by term. The linearity of integration is also a relevant property in this context because it allows the integral of a sum of functions to be split into the sum of their integrals. However, the specific operation in question—integrating polynomial functions—directly invokes the power rule for the actual integration of \(x^n\) terms. While the product and chain rules are essential for differentiation and relate to how functions combine or affect each other during differentiation, they are not applicable in the straightforward integration

When integrating a polynomial function, the power rule of integration is utilized. This rule states that if you have a term in the form of (x^n), where (n) is a real number, the integral is given by:

[

\int x^n , dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)

]

This means that when you integrate a polynomial, each term (ax^n) in the polynomial can be individually integrated using this rule. The coefficient (a) remains the same, and you increase the exponent by one while dividing by the new exponent. This systematic approach allows for straightforward integration of polynomial expressions term by term.

The linearity of integration is also a relevant property in this context because it allows the integral of a sum of functions to be split into the sum of their integrals. However, the specific operation in question—integrating polynomial functions—directly invokes the power rule for the actual integration of (x^n) terms.

While the product and chain rules are essential for differentiation and relate to how functions combine or affect each other during differentiation, they are not applicable in the straightforward integration