Problem 4 - Calculus with Concepts in Calculus

Question

Find all antiderivatives of the given function. $$ -6 x+5 $$

Step-by-Step Solution

Answer

The antiderivative is \(-3x^2 + 5x + C\).

Step 1: Recall the Antiderivative Rules

To find the antiderivative (or indefinite integral) of a function, we apply the rules for antiderivatives. The primary rule for powers of x is: for any constant a and n ≠ -1, the antiderivative of \(a x^n\) is \( \frac{a}{n+1} x^{n+1} + C \), where C is the constant of integration.

Step 2: Integrate the Polynomial Term by Term

The given function is \(-6x + 5\). We will find the antiderivative by integrating each term separately.For \(-6x\), treat \(n = 1\) and it becomes \[ \int (-6x) \, dx = -6 \int x \, dx = -6 \left( \frac{x^{1+1}}{1+1} \right) = -6 \left( \frac{x^2}{2} \right) = -3x^2 \]For the constant term 5, recall that \[ \int a \, dx = ax + C \].Thus, \[ \int 5 \, dx = 5x + C \].

Step 3: Combine the Resulting Antiderivatives

Now, combine the antiderivatives of each term:The antiderivative of \(-6x + 5\) is the combination of the two results: \[-3x^2 + 5x + C \], where C is the constant of integration.

Key Concepts

Indefinite Integral

The concept of an indefinite integral is central in calculus, particularly in the study of antiderivatives. When we talk about the indefinite integral, we are referring to a set of functions whose derivatives yield a given function. This does not provide a single solution but rather a family of functions, which is why it is sometimes called the "general antiderivative."

• Symbolically, the indefinite integral of a function \( f(x) \) is written as \( \int f(x) \, dx \).
• The process of determining this set of functions is known as integration.
• The "\(+ C\)" at the end of the integrated function is crucial; it signifies the constant of integration, reflecting the infinite number of antiderivatives that differ only by a constant.

Understanding indefinite integrals is fundamental because they provide the foundational groundwork for the practice of integration in calculus, specifically for computing area under curves and solving differential equations.

Power Rule

The power rule is a key technique used when finding the antiderivative of terms in the form of polynomials. It's a straightforward rule that simplifies the integration process significantly. Here's how it works:

• For a term \( a x^n \) where \( a \) is a constant and \( n eq -1 \), the antiderivative is found using the formula: \( \frac{a}{n+1} x^{n+1} + C \).
• This rule essentially reverses the process of differentiation. When differentiating, we multiply by the power and decrease the power by one. In integration, we increase the power and divide by the new power.

For example, in the polynomial \(-6x\), applying the power rule involves increasing the power \(n = 1\) of \(x\) by 1 to get \(x^2\), then dividing \(-6\) by \(2\), giving \(-3x^2\). Each term of a polynomial is integrated separately and the results are combined to form the full antiderivative.

Constant of Integration

In calculus, the concept of the constant of integration, denoted as \( C \), is an essential part of solving indefinite integrals. Whenever you compute an indefinite integral, you're not just solving for one specific function, but uncovering an entire family of them:

• \( C \) represents any constant number since the derivative of a constant is zero, meaning any constant added to a function will not affect its derivative.
• This is why every indefinite integral includes a \(+ C\): it encompasses every possible antiderivative.

Consider the integration of a constant function like 5: the indefinite integral \( \int 5 \, dx \) results in \( 5x + C \). Here, \( C \) captures the notion that there are infinitely many functions, such as \( 5x + 1 \), \( 5x + 2 \), etc., all of which differentiate back to the constant 5. Failure to include \( C \) would overlook the complete solution set, excluding valid antiderivative functions.