Problem 5 - Calculus for Biology and Medicine

Question

Find the general antiderivative of the given function. $$ f(x)=(x-1)(x+1) $$

Step-by-Step Solution

Answer

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

Step 1: Expand the Expression

Begin by expanding the function \( f(x) = (x-1)(x+1) \). Use the distributive property (also known as the FOIL method for binomials) to expand it as follows:\[ (x-1)(x+1) = x^2 + x - x - 1 = x^2 - 1 \]

Step 2: Set Up the Integral

After expanding the expression, we now have \( f(x) = x^2 - 1 \). The next step is to set up the integral to find the antiderivative:\[ \int (x^2 - 1) \, dx \]

Step 3: Integrate Each Term Separately

Integrate each term in \( x^2 - 1 \) separately using the power rule of integration, which states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) where \( C \) is the constant of integration:\[ \int x^2 \, dx = \frac{x^{3}}{3} \]\[ \int 1 \, dx = x \]Putting it together:\[ \int (x^2 - 1) \, dx = \frac{x^3}{3} - x + C \]

Step 4: Formulate the General Antiderivative

Combine the results from the integration of each term to write the general antiderivative:\[ F(x) = \frac{x^3}{3} - x + C \]

Key Concepts

Integration

Integration is a fundamental concept in calculus, and it serves as the reverse process of differentiation. When we talk about finding an antiderivative, or "integrating" a function, we are essentially trying to find a new function whose derivative is the original function.

• The integral sign \( \int \) represents this process.
• The function we want to integrate is placed right after the integral sign, and it's usually followed by a differential (\( dx \), \( dy \), etc.) to denote the variable of integration.
• The goal is to find a function, usually denoted as \( F(x) \), that satisfies the condition \( F'(x) = f(x) \).

When we integrate a polynomial, like our example \( f(x) = x^2 - 1 \), we perform the integration on each term separately and then sum the results.Understanding integration is crucial because it allows us to evaluate areas under curves, solve differential equations, and model complex systems in various scientific fields.

Power Rule

The power rule is an essential tool in integration, especially when dealing with polynomials. It provides a direct method for integrating terms of the form \( x^n \), where \( n \) is any real number except \(-1\). The rule is straightforward to apply:

• For a term \( x^n \), the antiderivative is \( \frac{x^{n+1}}{n+1} \) plus a constant, \( C \).
• This makes calculating the integral of a polynomial a simple process of applying the rule to each term individually.

In our specific example, for integrating \( x^2 \):- According to the power rule, we calculate \( \int x^2 \, dx = \frac{x^3}{3} \).- The term \( -1 \) is a special case where the exponent is zero (since \( x^0 = 1 \)), so \( \int 1 \, dx = x \).This rule is particularly useful because it transforms integration into a set of easy-to-handle algebraic manipulations, allowing us to solve broad classes of integrals efficiently.

Polynomials

Polynomials are algebraic expressions consisting of variables and coefficients, structured in terms that are added or subtracted, each term comprising a variable raised to a power. They're very common in mathematics due to their simple yet versatile structure.

• Polynomials are written in the form \( a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \), where \( n \) is a non-negative integer.
• Each term comprises a coefficient \( a_i \) and a variable raised to some power.
• The degree of the polynomial is determined by the highest power of the variable present.

In the context of calculus and integration, polynomials are favored due to their straightforward nature. For example, our function \( f(x) = x^2 - 1 \) is a polynomial. Its simplicity allows us to apply rules like the power rule easily, making the integration process more manageable.Working with polynomials helps in developing a strong foundation in calculus, as many complex functions can often be approximated or broken down into polynomial terms, facilitating easier analysis and solution.