Problem 27 - Applied Calculus

Question

Find an antiderivative. $$p(x)=x^{2}-6 x+17$$

Step-by-Step Solution

Answer

The antiderivative is \( P(x) = \frac{x^3}{3} - 3x^2 + 17x + C \).

Step 1: Understand the Task

The task is to find an antiderivative of the given function \( p(x) = x^2 - 6x + 17 \). An antiderivative, also known as an indefinite integral, represents a function whose derivative is equal to the given function.

Step 2: Apply the Power Rule for Integration

To find the antiderivative of the polynomial \( p(x) = x^2 - 6x + 17 \), apply the power rule of integration. This rule states that for \( x^n \), the antiderivative is \( \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration.

Step 3: Integrate Each Term Separately

Integrate each term of \( p(x) \) separately:- The antiderivative of \( x^2 \) is \( \frac{x^3}{3} \).- The antiderivative of \( -6x \) is \( -3x^2 \) (since \( -6 \cdot \frac{x^2}{2} = -3x^2 \)).- The antiderivative of \( 17 \) is \( 17x \) (since \( 17 \cdot x \)).

Step 4: Combine the Integrated Terms and Add Constant

Combine the results from Step 3 and add the constant of integration \( C \). Thus, the antiderivative is:\[ P(x) = \frac{x^3}{3} - 3x^2 + 17x + C \]

Key Concepts

Indefinite Integral

An indefinite integral, also known as an antiderivative, is a concept where we seek to find a function whose derivative matches a given function. When you perform an indefinite integration, you are reversing the process of differentiation.
The resulting function is usually expressed in terms of a constant of integration, denoted typically as "C." This constant arises because when you differentiate a constant, the result is zero, meaning it has no effect on the original function. Thus, the indefinite integral of a function is represented as:

• \( \int f(x) \, dx \)
• The result is \( F(x) + C \), where \( F'(x) = f(x) \)

The indefinite integral tells us a family of functions rather than a single function. This is why the constant "C" is important, as it accounts for any function that differs by a constant having the same derivative.

Power Rule for Integration

The power rule for integration is a fundamental tool used in finding antiderivatives, especially when dealing with functions in the form of polynomials. It simplifies the integration process by providing a straightforward formula.
If you have a polynomial term \( x^n \), where \( n eq -1 \), the power rule states that:

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

This rule is "reverse" of the derivative power rule, which means it essentially undoes the differentiation.
For example, if you start with \( x^2 \), applying the power rule results in the antiderivative \( \frac{x^{3}}{3} + C \). It is crucial to remember to always add the constant of integration \( C \) after applying the power rule.

Polynomial Integration

Polynomial integration is an application of the indefinite integral and the power rule. It involves integrating each term of a polynomial separately and then summing the results.
For our example, the polynomial \( p(x) = x^2 - 6x + 17 \) can be broken down into separate terms, each of which is integrated using the power rule:

• The term \( x^2 \) integrates to \( \frac{x^3}{3} \).
• The term \( -6x \) integrates to \( -3x^2 \) since \( \int -6x \, dx = -6 \cdot \frac{x^2}{2} = -3x^2 \).
• The constant 17 integrates to \( 17x \), remembering that the indefinite integral of a constant \( a \) is \( ax + C \).

After integrating every term, combine the results: \( \frac{x^3}{3} - 3x^2 + 17x + C \).
This process exemplifies how polynomial integration operates, allowing us to draw antiderivatives and uncover general solution families for equations.