Problem 11 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the general antiderivative \(F(x)+C\) for each of the following. $$ f(x)=4 x^{5}-x^{3} $$

Step-by-Step Solution

Answer

The general antiderivative is \( F(x) = \frac{2}{3}x^6 - \frac{1}{4}x^4 + C \).

Step 1: Identify the Function

First, we identify the given function which is this: \[ f(x) = 4x^5 - x^3 \]

Step 2: Apply the Power Rule for Antiderivatives

To find the antiderivative, apply the power rule: For any function of the form \[ ax^n \]its antiderivative is given by \[ \frac{a}{n+1}x^{n+1} \]

Step 3: Antiderivative of the First Term

For the term \( 4x^5 \), apply the power rule:\[F_1(x) = \frac{4}{5+1}x^{5+1} = \frac{4}{6}x^6 = \frac{2}{3}x^6\]

Step 4: Antiderivative of the Second Term

For the term \( -x^3 \), apply the power rule:\[F_2(x) = \frac{-1}{3+1}x^{3+1} = \frac{-1}{4}x^4 = -\frac{1}{4}x^4\]

Step 5: Combine and Add Constant of Integration

Add the antiderivatives from each term and include the constant of integration \( C \):\[F(x) = \frac{2}{3}x^6 - \frac{1}{4}x^4 + C\]

Key Concepts

Understanding the Power Rule

When we talk about antiderivatives, one of the most frequently used techniques is the power rule. It's simple yet powerful. If you have a function in the form of \( ax^n \), the antiderivative is found by adding one to the exponent and then dividing the coefficient by this new exponent value. In mathematical terms, when you have \( ax^n \), the antiderivative becomes \( \frac{a}{n+1}x^{n+1} \).
This rule transforms complex polynomial functions into something much more manageable. However, it's essential to pay attention to each term in the expression separately.
Remember to apply the rule to one term at a time and keep track of the signs and coefficients throughout the process.

The Importance of the Integration Constant

The phrase "plus C" that accompanies the general antiderivative is more than just a technicality. This \( C \) represents the "constant of integration". What the constant signifies is the infinite number of potential solutions when finding an antiderivative.
During differentiation, any constant disappears. That means when we reverse the process by finding the antiderivative, we should account for all those lost constants.

• "C" accounts for any constant that could have been present in the original function.
• It's a crucial part of accurately representing the general form of an antiderivative.
• Without adding \( C \), the solution is incomplete and may lead to incorrect applications.

The next time you see "+ C", remember that it's there to ensure that every possible original function is included.

Working with Polynomial Functions

Polynomial functions, like the one in the exercise, are expressions that involve variables raised to various powers combined with coefficients. They are the cornerstone of algebra and calculus operations.
The beauty of polynomial functions lies in their simplicity and predictability when it comes to differentiation and integration.
When dealing with polynomials:

• Each term is handled independently, meaning you find the antiderivative for each one separately.
• The power rule shines here, making it straightforward to find antiderivatives.
• Always remember to manage negative signs and coefficients carefully to avoid errors.

Incorporating these techniques allows you to handle even more complex polynomial expressions with ease, sharpening your mathematical skills across various disciplines.