Problem 2 - Calculus

Question

Find the general antiderivative \(F(x)+C\) for each of the following. $$ f(x)=x-4 $$

Step-by-Step Solution

Answer

The antiderivative is \( F(x) = \frac{x^2}{2} - 4x + C \).

Step 1: Identify the Function to Integrate

The function we need to find the antiderivative for is given as \( f(x) = x - 4 \). This is a linear function, and we will find its antiderivative term by term.

Step 2: Apply the Power Rule to the Term \(x\)

The power rule for integration states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) for \( n eq -1 \). Apply this to the term \( x = x^1 \). This gives us: \( \frac{x^{1+1}}{1+1} = \frac{x^2}{2} \).

Step 3: Integrate the Constant Term \(-4\)

The antiderivative of a constant \( c \) is \( cx \). So, the antiderivative of \(-4\) is \(-4x\).

Step 4: Combine Results

Combine the results from Steps 2 and 3 to find the antiderivative of \( f(x) \). Thus, \( F(x) = \frac{x^2}{2} - 4x + C \), where \( C \) is the constant of integration.

Key Concepts

power rule for integration

When tackling problems that involve finding the antiderivative of a function, one essential tool is the "power rule for integration." This rule is a method applied to integrate functions of the form \( x^n \), where \( n \) is any real number that is not -1. The power rule states:

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

This formula tells us that to find the antiderivative, you increase the exponent by one and then divide by the new exponent.For example, if you're integrating \( x^3 \), increase the exponent to 4, then divide \( x^4 \) by 4, resulting in \( \frac{x^4}{4} + C \). Remember, this rule only applies when \( n eq -1 \). For \( n = -1 \), the antiderivative is \( \ln{|x|} + C \). This power rule helps break down functions into simpler parts for easier integration.

linear function

A "linear function" is one of the simplest forms of a function and can generally be written as \( f(x) = ax + b \), where \( a \) and \( b \) are constants. The graph of a linear function is a straight line, and it has a constant slope \( a \). In our exercise, the linear function given is \( f(x) = x - 4 \). Here, \( a = 1 \) and \( b = -4 \). It's important to note that when finding the antiderivative of a linear function, you can integrate each term separately:

• The term \( ax \) uses the power rule for integration, turning into \( \frac{x^{n+1}}{n+1} \).
• The constant term \( b \) integrates into \( bx \).

Separating and integrating each term of the linear function helps simplify the process and leads to the correct antiderivative.

constant of integration

The "constant of integration," often denoted as \( C \), is an integral component of indefinite integrals. Whenever we calculate an indefinite integral or an antiderivative, we add this constant. Why do we add \( C \)? Since differentiation wipes out any constant terms (as the derivative of a constant is zero), the reverse process of integration does not uniquely determine a single function. All antiderivatives differ by a constant term. Adding \( C \) accounts for that unknown constant.In our example, once we integrated the terms \( x \) and \(-4\), we obtained \( \frac{x^2}{2} - 4x \). To complete the antiderivative, we include \( + C \) at the end. Thus, every indefinite integral has this constant, reminding us of all the possible vertical shifts of the antiderivative function. In summary, the constant of integration is crucial for making sure every possible function derived by integration from a derivative is accounted for.