Problem 3 - Calculus

Question

Find the general antiderivative \(F(x)+C\) for each of the following. $$ f(x)=x^{2}+\pi $$

Step-by-Step Solution

Answer

The general antiderivative is \( F(x) = \frac{1}{3}x^3 + \pi x + C \).

Step 1: Identify the function components

The function given is \( f(x) = x^2 + \pi \). We observe two components in this function: the polynomial term \( x^2 \) and the constant term \( \pi \).

Step 2: Find the antiderivative of the polynomial term

The antiderivative of a polynomial term \( ax^n \) is found using the formula \( \frac{a}{n+1}x^{n+1} + C \). Applying this to \( x^2 \), we have \( \frac{1}{2+1}x^{2+1} = \frac{1}{3}x^3 \).

Step 3: Find the antiderivative of the constant term

The antiderivative of a constant \( c \) is simply \( cx + C \). Therefore, the antiderivative of \( \pi \) is \( \pi x \).

Step 4: Combine the antiderivatives

Combine the antiderivative of each component to get the general antiderivative of the entire function. Thus, we have \( F(x) = \frac{1}{3}x^3 + \pi x + C \).

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers of the variable. Each power has its coefficient. Consider the function given in the original problem: \(f(x) = x^2 + \pi\). This is a simple polynomial because it involves a single variable raised to a power, along with constants like \(\pi\). Polynomial functions can come in more complex forms with multiple terms where each is a product of a constant and a variable raised to a power.

• The degree of a polynomial is determined by the highest power of the variable. For \(x^2\), it is 2.
• Polynomials can have more than one term. For example, \(2x^3 + 3x^2 + x + 5\) has four terms.
• They are easy to work with when finding antiderivatives because each term can be integrated separately.

Understanding polynomials is crucial in calculus, as they form the foundation for exploring more complex functions.

Constant Functions

Constant functions are among the simplest types of mathematical functions. A constant function has the same value regardless of what variable is plugged in. It does not depend on the variable at all. An example in the exercise is the constant \(\pi\).

• In a mathematical expression, constant terms provide fixed values that do not change.
• The graph of a constant function is a horizontal line.
• When integrating, the antiderivative of a constant \(c\) is \(cx\) plus a constant of integration \(C\).

In the given problem, the constant \(\pi\) contributes \(\pi x\) to the antiderivative, reflecting how constants translate to linear functions after integration. This principle is part of why calculus allows prediction of changes over time or space.

Integration Techniques

Integration techniques are methods used to find the antiderivative, or the 'reverse' of differentiation. When you integrate a function, you are essentially finding a new function whose derivative is the original function you started with. Several techniques exist for integration, but in this problem, we focus on simple, common types used for polynomials and constants.

• Power Rule for Integration: For any term \(ax^n\), the antiderivative is \(\frac{a}{n+1}x^{n+1}\).
• Constant Rule: The antiderivative of a constant \(c\) is \(cx + C\).

These rules apply here to turn \(x^2\) into \(\frac{1}{3}x^3\) and \(\pi\) into \(\pi x\). Understanding integration techniques is key to unraveling complex calculus problems, allowing us to reconstruct original functions from their rates of change.