Problem 1 - Calculus for Biology and Medicine

Question

Find the general antiderivative of the given function. $$ f(x)=x^{2}-4 x $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

We are tasked with finding the general antiderivative, which is the most general form of the inverse operation of differentiation, for the function \( f(x) = x^2 - 4x \). This involves finding a function \( F(x) \) such that \( F'(x) = f(x) \).

Step 2: Apply the Antiderivative Rules

Recall that the antiderivative of \( x^n \) (where \( n eq -1 \)) is \( \frac{x^{n+1}}{n+1} \), and for a constant \( c \), it is \( cx \). We'll apply these rules to each term in the function.

Step 3: Integrate Term by Term

The function is \( f(x) = x^2 - 4x \). We integrate each term separately:1. \( \int x^2 \, dx = \frac{x^{2+1}}{2+1} = \frac{x^3}{3} \).2. \( \int -4x \, dx = -4 \int x \, dx = -4 \cdot \frac{x^{1+1}}{1+1} = -4 \cdot \frac{x^2}{2} = -2x^2 \).

Step 4: Combine the Integrated Terms and Add Constant

Combine the integrated terms and include the constant of integration \( C \) to express the general antiderivative:\[ F(x) = \frac{x^3}{3} - 2x^2 + C \], where \( C \) is an arbitrary constant.

Key Concepts

Understanding Calculus and Its Role in Antiderivatives

Calculus is a branch of mathematics that studies change. It is fundamentally divided into differential calculus and integral calculus. While differential calculus focuses on the concept of the derivative, which measures how a function's value changes as its input changes, integral calculus revolves around the notion of the antiderivative or integral. An antiderivative is essentially a reverse process of differentiation. If you differentiate a function and then find its antiderivative, you return to the original function plus a constant.
When you have a function like \( f(x) = x^2 - 4x \), finding its general antiderivative means determining a function \( F(x) \), such that the derivative \( F'(x) \) equals \( f(x) \). In this context, calculus allows us to evaluate and handle continuous change, which is crucial in fields ranging from physics to engineering.
In finding the antiderivative of a polynomial function, such as \( f(x) = x^2 - 4x \), the process involves applying integration rules to each term individually, emphasizing calculus's versatile and systematic approach.

Integration Rules in Finding Antiderivatives

Integration rules provide the guidelines for finding the antiderivative of a function. These rules help transform the \( f(x) \) expression into the antiderivative \( F(x) \).
For any power of \( x^n \), as long as \( n eq -1 \), the rule to find the antiderivative is \( \int x^n \, dx = \frac{x^{n+1}}{n+1} \). This is the power rule for integration and is pivotal in solving polynomial functions. Some other rules of integration include:

• The integral of a constant \( c \) is \( cx \).
• The integral of a sum of functions is the sum of their integrals.
• The constant multiple rule, which states \( \int c \cdot f(x) \, dx = c \cdot \int f(x) \, dx \).

For the function \( f(x) = x^2 - 4x \):

• Apply the power rule to \( x^2 \): \( \int x^2 \, dx = \frac{x^3}{3} \).
• Use the constant multiple rule for \( -4x \): \( -4 \int x \, dx = -4 \cdot \frac{x^2}{2} = -2x^2 \).

Integration rules are fundamental, allowing students to break down components of complex expressions into simpler, manageable parts.

The Constant of Integration Explained

The constant of integration, denoted by \( C \), is a crucial aspect of finding antiderivatives. When calculating an indefinite integral, you determine a family of functions that differ by a constant. That constant represents all potential vertical shifts of the antiderivative \( F(x) \). This happens because taking the derivative of a constant results in zero; therefore, any constant added to \( F(x) \) will not affect \( F'(x) \).
For the problem at hand, where \( f(x) = x^2 - 4x \), after integrating, we get:\[ F(x) = \frac{x^3}{3} - 2x^2 + C \]The inclusion of \( C \) signifies that without additional information, such as an initial condition, there are infinitely many functions that can serve as antiderivatives.
This constant becomes especially relevant in solving differential equations and real-world problems where specific constraints help determine the value of \( C \). For students, remembering to add \( C \) ensures that their solution to the antiderivative is valid and complete.